Problems in-plane-geometry-Sharygin

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I.F. Sharygin

Problems in Plane Geometry

Mir
Publishers
Moscow

Translated from Russian
by Leonid Levant

First published 1988
Revised from the 1986 Russian edition

m17020'0000-304
056(01)-88

21-88

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@ English translation,

Mir Publishers, 1988
ISBN 5-03-000180-8

Contents

Preface to the English Edition

.

Section t. Fundamental Geometrical Facts
and Theorems. Computational
Problems . . .
Section 2. Selected Problems and Theorems of Plane Geometry 65. Carnot's Theorem 65. Ceva's and
Menelaus' Theorems. Affine Problems 70. Loci of Points 82.
Triangles. A Triangle and a
Circle 89. Quadrilaterals t t 7.
Circles aod Tangents. Feuerbach's Theorem t29. Combina-

6

8

tions of Figures. Displacements

in the Plane. Polygons t37. Geometrical Inequalities. Problems
on Extrema 147
Answers, Hint!, Solutions
Section t
Section 2

.

.

too

too

.

214

Appendix. Inversion

392

Preface to the English Edition
This is a translation from the revised
edition of the Russian book which was
issued in 1982. It is- actually the first in
a two-volume work on solving problems in
geometry, the second volume "Problems in
Solid Geometry" having been published in
English first by Mir Publishers in ~986.
Both volumes are designed for schoolchildren and teachers.
This volume contains over 600 problems
in plane geometry and" consists of two
parts. The first part contains rather simple
problems to be solved in classes and at
home. The second part also contains hints
and detailed solutions. Over ~OO new problems have been added to the 1982 edition,
the simpler problems in the first addition
having been eliminated, and a number of
new sections- (circles and tangents, polygons, combinations of figures, etc.) having
been introduced, The general structure of
the book has been changed somewhat to
accord with the new, more detailed, classification of the problems. As a result, all
the problems in this volume have been
rearranged.
Although the problems in this collection
vary in "age" (some of them can be found
in old books and journals, others were
offered at mathematical olympiads or published in the journal "Quant" (Moscow»,
I still hope that some of the problems in

Preface

this collection will be of interest to experienced geometers.
Almost every geometrical problem is nonstandard (as compared with routine exercises on solving equations, inequalities,
etc.): one has to think of what additional
constructions must be made, or which formulas and theorems must be used. Therefore, this collection cannot be regarded as
a problem-book in geometry; it is rather a
collection of geometrical puzzles aimed at
demonstrating the elegance of elementary
geometrical techniques of proof and methods
of computation (without using vector algebra and with a minimal use of the method
of coordinates, geometrical transformstions, though a somewhat wider use of trigonometry).
In conclusion, I should like to thank
A.Z. Bershtein who assisted me in preparing the first section of the book for print.
I am also grateful to A.A. Yagubiants who
let me know several elegant geometrical
facts.

The Author

Section 1

Fundamental Geometrical Facts
and Theorems.
Computational Problems
t. Prove that the medians in a triangle
intersect at one point (the median point)
and are divided by this point in the ratio
1 : 2.
•2. Prove that the medians separate the
triangle into six equivalent parts.
3. Prove that the diameter of the circle
circumscribed about a triangle is equal
to the ratio of its side to the sine of the
opposite angle.
4. Let the vertex of an angle be located
outside a circle, and let the sides of the
angle intersect the circle. Prove that the
angle is measured by the half-difference of
the arcs inside the angle which are cut out
by its sides on the circle.
5. Let the vertex of an angle lie inside
a circle. Prove that the angle is measured
by the half-sum of the arcs one of which
is enclosed between its sides and the other
between their extensions.
6. Let AB denote a chord of a circle, and
l the tangent to the circle at the point A.
Prove that either of the two angles between
AB and l is measured by the half-arc of the

Sec. f. Fundamental Facts

9

circle enclosed inside the angle under consideration.
7. Through the point M located at a distance a from the centre of a circle of radius R
(a > R), a secant is drawn intersecting the
circle at points A and B. Prove that the
product I MA 1·1 MB J is constant for all
the secants and equals a2 - R2 (which is
the squared length of the tangent).
8. A chord AB is drawn through the point
M situated at a distance a from the centre
of a circle of radius R (a < R). Prove that
I AM I-I MB I is constant for all the chords
and equals RI - a2 •
9. Let AM be an angle bisector in the
triangle ABC. Prove that IBM I: I CMf =
I AB I I AC I. The same is true for
the bisector of the exterior angle of the
triangle. (In this case the .point M lies on the
extension of the side Be.)
10. Prove that the sum of the squares of
the lengths of the diagonals of a parallelogram is equal to the sum of the squares of
the lengths of its sides.
1t. Given the sides of a triangle (a, b,
and e). Prove that the median mel drawn to
the side a can be computed by the formula
ma

= ~ V2lJ2+ 2c2 - a2 •

12. Given two triangles having one vertex A in common, the other vertices being
situated on two straight lines passing

10


through A. Prove that the ratio of the areas
of these triangles is equal to the ratio of
the products of the two sides of each triangle
emanating from the vertex A.
13. Prove that the area of the circumscribed polygon is equal to rp, where r
is the radius of the inscribed circle and p
its half-perimeter (in particular, this formula holds true for a triangle).
14. Prove that the area of a quadrilateral
is equal to half the product of its diagonals'
and the sine of the angle between them. '15. Prove the validity of the following
formulas for the area of a triangle:
S=

a l sin B sin C
2...
2. A ' S = 2R SID A SID B sm C,
sm

where A, B, C are its angles, a is the side
lying opposite the angle A, and R is the
radius of the circumscribed circle.
16. Prove that the radius of the circle
inscribed in a right triangle can be coma+b-c
puted by the formula r =
2
'
where a and b are the legs and c is the
hypotenuse.
17. Prove that if a and b are two sides of
a triangle, a the angle between them, and l
the bisector of this angle, then
(£

2abc08

1=

a+b

T

Sec. 1. Fundamental Facts

it

18. Prove that the distances from the
vertex A of the triangle ABC to the points
of tangency of the inscribed circle with the
sides AB and AC are equal to p - a (each),
where p is the half-perimeter of the triangle
ABC, a = I BC I.
19. Prove that if in a convex quadrilateral ABeD I AB I + I CD I = I AD I
I BC I, then there is a circle touching all
of its sides.
20. (a) Prove that the altitudes in a
triangle 'are concurrent (that is intersect at
one point). (b) Prove that the distance
from any vertex of a triangle to the point
of intersection of the altitudes is twice
the distance from the centre of the circumscribed circle to the opposite side.

+

• • •
21. Points A and B are taken on one side
of a right angle with vertex 0 and lOA I =
a, I OB I = b. Find the radius of the circle
passing through the points A and Band
touching the other side of the angle.
22. The hypotenuse of a right triangle is
.equal to c, one of the acute angles being
30°. Find the radius of the circle with
centre at the vertex of the angle of 30°
which separates the triangle into two equi valent parts.
23. The legs of a right triangle are a and
b. Find the distance from the vertex of the

t2


right angle to the nearest point of the
inscribed circle.
24. One of the medians of a right triangle
is equal to m and divides the right angle
in the ratio 1 : 2. Find the area of the
triangle.
25. Given in a triangle ABC are three
sides: I Be I = a, I CA I = b, I AB I = c.
Find the ratio in which the point of intersection of the angle bisectors divides the
bisector of the angle B.
26. Prove that the sum of the distances
from any point of the base of an isosceles
triangle to its sides is equal to the altitude
drawn to either of the sides.
1:1. Prove that the sum of distances from
any point inside an equilateral triangle
to its sides is equal to the altitude of this
triangle.
28. In an isosceles triangle ABC, taken
on the base A C is a point M such that
I AM I = a, I Me I = b. Circles are inscribed in the triangles ABM and CBM.
Find the distance between the points at
which these circles touch the side BM.
29. Find the area of the quadrilateral
bounded by the angle bisectors of a parallelogram with sides a and b and angle Ct.
30. A circle is inscribed in a rhombus
with altitude h and acute angle C%. Find
the radius of the greatest of two possible
circles each of which touches the given
circle and two sides of the rhombus.


13

3t . Determine the acute angle of the
rhombus whose side is the geometric mean
of its diagonals.
32. The diagonals of a convex quadrilateral are equal to a and b, the line segments
joining the midpoints of the opposite sides
are congruent. Find the area of the quadrilateral.
33. The side AD of the rectangle ABCD
is three times the side AB; points M and
N divide AD into three equal parts. Find
LAMB + LANB + LADB.
34. Two circles intersect at points A
and B. Chords A C and AD touching the
given circles are drawn through the point
2.1
A. Prove that lAC 1 BD I = I AD 12 •
IBC

I.

35. Prove that the bisector of the right
angle in a right triangle bisects the angle
between the median and the altitude drawn
to the hypotenuse.
36. On a circle of radius r, three points
are chosen 80 that the circle is divided
into three arcs in the ratio 3 : 4 : 5. At the
division points, tangents are drawn to the
circle. Find the area of the triangle formed
by the tangents.
37. An equilateral trapezoid is circumscribed about a circle, the lateral side of
the trapezoid is I, one of its bases is equal
to 4. Find the area of the trapezoid.
38. Two straight lines parallel to the
bases of a trapezoid divide each lateral

14


side into three equal parts. The entire
trapezoid is separated by the lines into
three parts. Find the area of the middle
part if the areas of the upper and lower
parts are 8 1 and S2' respectively.
39. In the trapezoid ABCD I AB I = a,
I BC I = b (a =1= b). The bisector of the
angle A intersects either the base BC or the
lateral side CD. Find out which of them?
40. Find the length of the line segment
parallel to the bases of a trapezoid and
passing through the point of intersection
of its diagonals if the bases of the trapezoid
are a and b.
41. In an equilateral trapezoid circumscribed about 8 circle, the ratio of the
parallel sides is k. Find the angle at the
base.
42. In a trapezoid ABCD, the base AB
is equal to a, and the base CD to b. .F ind
the area of the trapezoid if the diagonals
of the trapezoid are known to be the bisectors of the angles DAB and ABC.
43. In an equilateral trapezoid, the midline is equal to a, and the diagonals are
mutually perpendicular. Find the area of
the trapezoid.
44. The area of an equilateral trapezoid
circumscribed about a circle is equal to S,
and the altitude of the trapezoid is half
its lateral side. Determine the radius of
the circle inscribed in the trapezoid.
45. The areas of the triangles formed by

i5

the segments of the diagonals of a trapezoid
and its bases are equal to 8 1 and 8 1 • Find
the area of the trapezoid.
46. In a triangle ABC, the angle ABC is
tX. Find the angle AOe, where 0 is the
centre of the inscribed circle.
47. The bisector of the right angle is
drawn in a right triangle. Find the distance
between the points of intersection of the
altitudes of the triangles thus obtained,
if the legs of the given triangle are a and b.
48. A straight line perpendicular to two
sides of a parallelogram divides the latter
into two trapezoids in each of which a
circle can .be inscribed. Find the acute
angle of the parallelogram if its sides are
a and b (a < b).
.
49. Given a half-disc with diameter AB.
Two straight lines are drawn through the
midpoint of the semicircle which divide
the half-disc into three equivalent areas.
In what ratio is the diameter AB divided
by these lines?
50. A square ABCD with side a and two
circles are constructed. The first circle is
entirely inside the square touching the side
AB at a point E and also the side Be and
diagonal AC. The second circle with centre
at A passes through the point E. Find
the area of the common part of the two
discs bounded by these circles.
51. The vertices of a regular hexagon
with side a are the centres of the circles

16


with radius a/y'2. Find the area of the
part of the hexagon not enclosed by these
circles.
52. A point A is taken outslde a circle
of radius R. Two secants are drawn from
this point: one passes through the centre,
the other at a distance of R/2 from the
centre. Find the area of the region enclosed
between these secants.
53. In a quadrilateral ABeD: LDAB =
90°, LDBC = 90°. I DB I = a, and I DC 1=
b. Find the distance between the centres
of two circles one of which passes through
the points D, A and B, the other through
the points B, C, and D.
54. On the sides AB and AD of the
rhombus ABCD points M and N are taken
such that the straight lines Me and NC
separate the rhombus into three equivalent
parts. Find I MN I if I BD I = d.
55. Points M and N are taken on the
side AB of a triangle ABC such that
lAM I: I MN I: INB I = 1: 2 : 3.',Through
the points M and N straight Iines are
drawn parallel to tHe side A C. Find the
area of the part of the triangle enclosed
between these lines if the area of the triangle
ABC is equal to S.
56. Given a circle and a point A located
outside of this circle, straight lines AB
and AC are tangent to it (B and C points
of tangency). Prove that the centre of the

Sec. t. Fundamental Facts

i7

circle inscribed in the triangle ABC lies
on the given circle.
57. A circle is circumscribed about an
equilateral triangle ABC, and an arbitrary
point M is taken on the arc BC. Prove that

I AM I = I EM I + I CM I·

58. Let H be the point of intersection of
the altitudes in a triangle ABC. Find the
interior angles of the triangle ABC if
LBAH = a, LABH = ~.
59. The area of a rhombus is equal to S,
the sum of its diagonals is m, Find the side
of the rhombus.
60. A square with side a is inscribed in
a circle. Find the side of the square inscribed in one of the segments thus obtained.
61. In a 1200 segment of a circle with
altitude h a rectangle ABCD is inscribed
so that I AB I : I BC I = 1 4 (BC lies
on the chord). Find the area of the rectangle.
62. The area of an annulus is equal to S.
The radius of the larger circle is equal to
the circumference of the smaller. Find the
radius of the smaller circle.
63. Express the side of a regular decagon
in terms of the radius R of the circumscribed
circle.
64. Tangents MA and MB are drawn
from an exterior point M to a circle of
radius R forming an angle a. Determine
2-01557

18


the area of the figure bounded by the tangents and the minor arc of the circle.
65. Given a square ABCD with side a.
Find the centre of the circle passing
through the following points: the midpoint
of the side AB, the centre of the square,
and the vertex C.
66. Given a rhombus with side a and acute
angle Ct. Find the radius of the circle passing through two neighbouring vertices of
the rhombus and touching the opposite
side of the rhombus or its extension.
67. Given three pairwise tangent circles
of radius r. Find the area of the triangle
formed by three lines each of which touches
two circles and does not intersect the third
one.
68. A circle of radius r touches a straight
line at a point M. Two, points A and B
are chosen on this line on opposite sides of
M such that I MAl = 1MB I = a. Find
the radius of the circle passing through A
and B and touching the given circle.
69. Given a square ABCD with side a.
Taken on the side BC is a point M such that
I BM I = 3 I MC I and on the side CD a
point N such that 2 I CN I = I ND I. Find
the radius of the circle inscribed in the
triangle AMN.
70. Given a square ABeD with side a.
Determine the distance between the midpoint of the line segment AM, where M is


19

the midpoint of Be, and a point N on the
side CD such that I CN I I ND I = 3 1.
71. A straight line emanating from the
vertex A in a triangle ABC bisects the
median BD (the point D lies on the side
AC). What is the ratio in which this line
divides the side BC?
72. In a right triangle ABC the leg CA
is equal to b, the leg eB is equal to a, CH
is the altitude, and AM is the median.
Find the area of the triangle BMH.
73. Given an isosceles triangle ABC whose
LA = a > 90° and I Be I = a. Find the
distance between the point of intersection
of the altitudes and the centre of the circumscribed circle.
74. A circle is circumscribed about a
triangle ABC where I BC I = a, LB = a,
LC = p. The bisector of the angle A meets
the circle at a point K. Find I AK I.
75. In a circle of radius R, a diameter is
drawn with a point A taken at a distance
a from the centre. Find the radius of another
circle which is tangent to the diameter at
the point A and touches internally the
given circle.
76. In a circle, three pairwise intersecting
chords are drawn. Each chord is divided
into three equal parts by the points of
intersection. Find the radius of the circle
if one of the chords is equal to a.
77. One regular hexagon is inscribed in
a circle, the other is circumscribed about
2*

20


it. Find the radius of the circle if the difference between the perimeters of these hexagons is equal to a.
78. In an equilateral triangle ABC whose
side is equal to a, the altitude BK is drawn.
A circle is inscribed in each of the triangles
ABK and BCK, and a common external
tangent, different from the side AC, is drawn
to them. Find the area of the triangle cut
off by this tangent from the triangle ABC.
79. Given in an inscribed quadrilateral
ABCD are the angles: LDAB = a, LABC=
p, LBKC = y, where K is the point
of intersection of the diagonals. Find the
angle ACD.
80. In an inscribed quadrilateral ABeD
whose diagonals intersect at a point K,
lAB I = a, 18K 1= b, IAK 1= c, I CDI=
d. Find I AC I.
8t. A circle is circumscribed about a
trapezoid. The angle between one of the
bases of the trapezoid and a lateral side is
equal to ex and the angle between this base
and one of the diagonals is equal to p.
Find the ratio of the area of the circle to
82. In an equilateral trapezoid ABCD,
the base AD is equal to a, the base Be
is equal to b, I AB I = d. Drawn through
the vertex B is a straight line bisecting the
diagonal A C and intersecting AD at a point
K. Find the area of the triangle BD K .
83. Find the sum of the squares of the


21

distances from the point M taken on a diameter of a circle to the end points of any
chord parallel to this diameter if the radius
of the circle is R, and the distance from M
to the centre of the circle is a.
84. A common chord of two intersecting
circles can be observed from their centres
at angles of 90° and 60°. Find the radii of
the circles if the distance between their
centres is equal to a.
85. Given a regular triangle ABC. A point
K di vides the side A C in the ratio 2 : 1,
and a point M divides the side AB in the
ratio t 2 (as measured from the vertex A
in both cases). Prove that the length of the
line segment KM is equal to the radius of
the circle circumscribed about the triangle

ABC.
86. Two circles of radii Rand R/2 touch
each other externally. One of the end points
of the line segment of length 2R forming
an angle of 30° with the centre line coincides
with the centre of the circle of the smaller
radius. What part of the line segment lies
outside both circles? (The line segment
intersects both circles.)
87. A median BK, an angle bisector BE,
and an altitude AD are drawn in a triangle
ABC. Find the side AC if it is known that
thelines EX and BE divide the line segment
AD into three equal parts and I AB I = 4.
88. The ratio of the radius of the circle
inscribed in an isosceles trrangle to the

22


radius of the circle circumscribed about this
triangle is equal to k. Find the base angle
of the triangle.
89. Find the cosine of the angle at the
base of an isosceles triangle if the point of
intersection of its altitudes lies on the circle
inscribed in the triangle.
90. Find the area of the pentagon bounded
by the lines BC, CD, AN, AM, and BD,
where A, B, and D are the vertices of a
square ABCD, N the midpoint of the side
BC, and M divides the side CD in the ratio
2 : 1 (counting from the vertex C) if the
side of the square ABCD is equal to a.
91. Given in a triangle ABC: LBAC =
a, LABC =~. A circle centred at B
passes through A and intersects the line AC
at a point K different from A, and the line
Be at points E and F. Find the angles of
the triangle E KF.
92. Given a square with side a. Find the
area of the regular triangle one of whose
vertices coincides with the midpoint of one
of the sides of the square, the other two
lying on the diagonals of the square.
93. Points M, N, and K are taken on the
sides of a square ABeD, where M is the
midpoint of AB, N lies on the side BC
(2 I BN I = I NC I). K lies on the side
DA (2 I DK I = I KA I). Find the sine
of the angle between the lines MC and N K.
94. A circle of radius r passes through the
vertices A and B of the triangle ABC and

23


intersects the side BC at a point D. Find
the radius of the circle passing through the
points A,D, and C if I AB 1= c, I AC I =b.
95. In a triangle ABC, the side AB is
equal to 3, and the altitude CD dropped
on the side AB is equal to V3: The foot D
of the altitude CD lies on the side AB, and
the line segment AD is equal to the side BC.
Find I AC I.
96. A regular hexagon ABCDEF is inscribed in a circle of radius R. Find ·the
radius of the circle inscribed in the triangle
ACD.

97. The side AB of a square ABCD is
equal to 1 and is a chord of a circle, the
rest of the sides of the square lying outside
this circle. The length of the tangent CK
drawn from the vertex C to the circle is
equal to 2. Find the diameter of the circle.
98. In a right triangle, the smaller angle
is equal to a. A straight line drawn perpendicularly to the hypotenuse divides
the triangle into two equivalent parts.
Determine the ratio in which this line
divides the hypotenuse.
99. Drawn inside a regular triangle with
side equal to 1 are two circles touching
each other. Each of the circles touches two
sides of the triangle (each side of the
triangle touches at least one of the circles).
Prove that the sum of the radii of these
circles is not less than
1}/2.

(va -

24


tOO. In a right triangle ABC with an
acute angle A equal to 30°, the bisector of
the other acute angle is drawn. Find the
distance between the centres of the two
circles inscribed in the triangles ABD and
CBD if the smaller leg is equal to t.
10t. In a trapezoid ABeD, the angles
A and D at the base AD are equal to 60°
and 30°, respectively. A point N lies on
the base Be, and I BN I : I NC I = 2.
A point M lies on the base AD; the straight
line M N is perpendicular to the bases of
the trapezoid and divides its area into
two equal parts. Find I AM I : I MD I.
102. Given in a triangle ABC: I Be I =
a, LA = a, LB = p. Find the radius of
the circle touching both the side A Cat
a point A and the side BC.
103. Given in a triangle ABC: I AB I =
c, IRG I = a, LB =~. On the side AR,
a point M is taken such that 2 I AM I =
3 I MB I. Find the distance from M to the
midpoint of the side AC.
t04. In a triangle ABC, a point M is taken
on the side AB and a point N on the side
AC such that I AM I = 3 I MB I and
2 I AN I = I NC I. Find the area of the
quadrilateral MBCN if the area of the
triangle ABC is equal to S.
t05. Given two concentric circles .of
radii Rand r (R > r) with a common centre
O. A third circle touches both of them.
Find the tangent of the angle between the

25

tangent lines to the third circle emanating
from the point O.
t06. Given in a parallelogram ABeD:
I AB I = a, I AD I = b (b > a), LBAD =
a (ex < 90°). On the sides AD and BC,
points K and M are taken such that BKDM
is a rhombus. Find the side of the rhombus.
107. In a right triangle, the hypotenuse
is equal to c. The centres of three circles
of radius cl5 are found at its vertices. Find
the radius of a fourth circle which touches
the three given circles and does not enclose
them.
108. Find the radius of the circle which
cuts on both sides of an angle ex chords of
length a if the distance between the nearest
end points of these chords is known to be
equal to b.
109. A circle is constructed on the side
BC of a triangle ABC as diameter. This
circle intersects the sides AB and AC at
points M and N, .respectively. Find the
area of the triangle AMN if the area of the
triangle ABC is equal to S, and LBAC=a.
110. In a circle of radius R two mutually
perpendicular chords M Nand PQ are
drawn. Find the distance between the points
M and P if I NQ I = a.
ttl. In a triangle ABC, on the largest
side BC equal to b, a point M is chosen.
Find the shortest distance between the
centres of the circles circumscribed about
the triangles BAM and A GM.
a'

26


112. Given in a parallelogram ABeD:

I AB I = a, I Be I = b, LABC =

ct. Find
the distance between the centres of the
circles circumscribed about the triangles
BCD and DAB.
113. In a triangle ABC, LA = a, I BA 1=
a, lAC 1 = b. On the sides AC and
AR, points M and N are taken, M being the
midpoint of AC. Find the length of the
line segment M N if the area of the triangle
AMN is 1/3 of the area of the triangle ABC.
114. Find the angles of a rhombus if the
area of the circle inscribed in it is half
the area of the rhombus.
tt5. Find the common area of two equal
squares of side a if one can be obtained
from the other by rotating through an angle
of 45° about its vertex.
116. In a quadrilateral inscribed in a
circle, two opposite sides are mutually
perpendicular, one of them being equal to
a, the adjacent acute angle is divided by
one of the diagonals into ct and ~. Determine
the diagonals of the quadrilateral (the
angle a is adjacent to the given side).
117. Given a parallelogram ABeD with
an acute angle DAB equal to a in which
I AB I = a, I AD I = b (a < b). Let K
denote the foot of the perpendicular dropped
from the vertex B on AD, and M the foot
of the perpendicular dropped from the point
K on the extension of the side CD. Find the
area of the triangle BKM.


27

118. In a triangle ABC, drawn from the
vertex C are two rays dividing the angle
ACB into three equal parts. Find the ratio
of the segments of these rays enclosed
inside the triangle if I BC I = 3 I AC I,
LACB = Ct.
119. In an isosceles triangle ABC (I AB 1=
I BC I) the angle bisector AD is drawn.
The areas of the triangles ABD and ADC
are equal to 8 1 and 8 2 , respectively. Find
lAC I.
120. A circle of radius R 1 is inscribed
in an angle Ct. Another circle of radius R 2
touches one of the sides of the angle at
the same point as the first one and intersects the other side of the angle at points
A and B. Find I AB I.
121. On a straight line passing through
the centre 0 of the circle of radius 12,
points A and B are taken such that 1 OA I =
15, I AB 1 = 5. From the points A and B,
tangents are drawn to the circle whose points
of tangency lie on one side of the line
OAB. Find the area of the triangle ABC,
where C is the point of intersection of these
tangents.
122. Given in a triangle ABC: I BC I =
a, LA = a, LB = p. Find the radius
of the circle intersecting all of i ts sides
and cutting off on each of them a chord of
length d.
123. In a convex quadrilateral, the line
segments joining the midpoints of the oppo-

28


site sides are equal to a and b and intersect
at an angle of 60° Find the diagonals of
the quadrilateral.
124. In a triangle ABC, taken on the
side Be is a point M such that the distance
from the vertex B to the centre of gravity
of the triangle AMC is equal to the distance
from the vertex C to the centre of gravity
of the triangle AMB. Prove that I BM I =
I DC I where D is the foot of the altitude
dropped from the vertex A to Be.
125. In a right triangle ABC, the bisector
BE of the right angle B is divided by the
centre 0 of the inscribed circle so that
I BO I I OE I =
Find· the acute
angles of the triangle.
126. A circle is constructed on a line
segment AB of length R as diameter. A second circle of the same radius is centred at
the point A. A third circle touches the
first circle internally and the second circle
externally; it also touches the line segment
AB. Find the radius of the third circle.
127. Given a triangle ABC. It is known
that I AB I = 4, I A C I = 2, and I BC I =
3. The bisector of the angle A intersects
the side BC at a point K. The straight line
passing through the point B and being
parallel to AC intersects the extension of
the angle bisector AK at the point M. Find

V3 V2.

IKMI·

128. A circle centred inside a right angle
touches one of the sides of the angle, inter-


29

sects the other side at points A and Band
intersects the bisector of the angle at points
C and D. The chord A B is equal to V~
the chord CD to V7. Find the radius of
the circle.
129. Two circles of radius 1 lie in a parallelogram, each circle touching the other
circle and three sides of the parallelogram.
One of the segments of the side from the
vertex to the point of tangency is equal
to V Find the area of the parallelogram.
130. A circle of radius R passes through
the vertices A and B of the triangle ABC
and touches the line A C at A. Find the
area of the triangle ABC if LB = a, LA =

a

~.

131. In a triangle ABC, the angle bisector
AK is perpendicular to the median BM,
and the angle B is equal to 120°. Find the
ratio of the area of the triangle A-BC to the
area of the circle circumscribed about this
triangle.
132. In a right triangle ABC, a circle
touching the side Be is drawn through the
midpoints of AB and AC. Find the part of
the hypotenuse A C which lies inside this
circle if I AB I = 3, I BC I = 4.
133. Given a line segment a. Three
circles of radius R are centred at the end
points and midpoint of the line segment.
Find the radius of the fourth circle which
touches the three given circles.

30


134. Find the angle between the common
external and internal tangents to two circles
of radii Rand r if the distance between
their centres equals -V 2 (R2 + r 2) (the centres of the circles are on the same side of the
common external tangent and on both sides
of the common internal tangent).
135. The line segment AB is the diameter
of a circle, and the point C lies outside
this circle. The line segments AC and BC
intersect the circle at points D and E,
respectively. Find the angle CBD if the
ratio of the areas of the triangles DCE and
ABC is 1 4.
136. In a rhombus ABCD of side a, the
angle at the vertex A is equal to 1200 •
Points E and F lie on the sides BC and AD,
respectively, the line segment EF and the
diagonal AC of the rhombus intersect at M.
The ratio of the areas of the quadrilaterals
REFA and ECDF is 1 : 2. Find I EM I
if I AM I I MC I = 1 3.
137. Given a circle of radius R centred
at O. A tangent AK is drawn to the circle
from the end point A of the line segment
VA, which meets the circle at M. Find
the radius of the circle touching the line
segments AK, AM, and the arc MK if
LOAK = 60°.
t38. Inscribed in a circle is an isosceles
triangle ABC in which I AB I = I Be I
and LB = p. The midline of the triangle


31

is extended to intersect the circle at points
D and E (DE II AC). Find the ratio of the
areas of the triangles ABC and pBE.
t39. Given an angle ex with vertex O.
A point M is taken on one of its sides and
a perpendicular is erected at this point
to intersect the other side of the angle at a
point N. Just in the same way, at a point
K taken on the other side of the angle a
perpendicular is erected to intersect the
first side at a point P. Let B denote the
point of intersection of the lines M Nand
KP, and A the point of intersection of the
lines OB and NB. Find lOA I if 10M I =
a and I OP I === b.
t40. Two circles of radii Rand r touch
the sides of a given angle and each other.
Find the radius of a third circle touching
the sides of the same angle and whose centre
is found at the point at which the given
circles touch each other.
t41. The distance between the centres
of two non-intersecting circles is equal to a.
Prove that the four points of intersection
of common external and internal tangents
lie on one circle. Find the radius of this
circle.
t42. Prove that the segment of a common
external tangent to two circles which is
enclosed between common internal tangents
is equal to the length of a common internal
tangent.
143. Two mutually perpendicular ra-

32


dii VA and OB are drawn in a circle
centred at O. A point C is on the arc AB
such that LAOe = 60° (LBOC = 30°).
A circle of radius AB centred at A intersects the extension of OC beyond the point
C at D. Prove that the line segment CD
is equal to the side of a regular decagon
inscribed in the circle.
Let us now take a point M diametrically
opposite to the point C. The line segment
MD, increased by 1/5 of its length, is assumed to be approximately equal to half the
circumference. Estimate the error of this
approximation.
144. Given a rectangle 7 X 8. One vertex
of a regular triangle coincides with one of
the vertices of the rectangle, the two other
vertices lying on its sides not containing
this vertex. Find the side of the regular
triangle.
145. Find the radius of the minimal circle
containing an equilateral trapezoid with
bases of 15 and 4 and lateral side of 9..
146. ABCD is a rectangle in which
I AB I = 9, I BC I = 7. A point M is
taken on the side CD such that I CM I =
3, and point N on the side AD such that
I AN I = 2.5. Find the greatest radius of
the circle which goes inside the pentagon
ABCMN.
147. Find the greatest angle of a triangle
if the radius of the circle inscribed in the
triangle with vertices at the feet of the


33

altitudes of the given triangle is half the
least altitude of the given triangle.
148. In a triangle ABC, the bisector of
the angle C is perpendicular to the median
emanating from the vertex B. The centre
of the inscribed circle lies on the circle
passing through the points A and C and the
centre of the circumscribed circle. Find
I AB I if I BC I = 1.
149. A point M is at distances of 2, 3 and
6 from the sides of a regular triangle (that
is, from the lines on which its sides are
situated). Find the side of the regular
triangle if its area is less than 14.
150. A point M is at distances -of
and 3
from the sides of an angle of 60°
(the feet of the perpendiculars dropped from
M on the sides of the angle lie on the sides
themselves, but not on their extensions).
A straight line passing through the point M
intersects the sides of the angle and cuts off
a triangle whose perimeter is 12. Find the
area of this triangle.
151. Given a rectangle ABeD in which
I AB I = 4, I Be I = 3. Find the side of
the rhombus one vertex of which coincides
with A, and three others lie on the line
segments AB, BC and BD (one vertex on
each segment).
152. Given a square ABCD with a side
equal to 1. Find the side of the rhombus one
vertex of which coincides with A, the oppo-

va

3-01557

va

34

'Problems in Plane Geometry

site vertex lies on the line BD, and the two
remaining vertices on the lines BC and CD.
153. In a parallelogram ABCD the acute
angle is equal to cx,. A circle of radius r
passes through the vertices A, B, and C
and intersects the lines AD and CD at points
M and N. Find the area of the triangle
BMN.
154. A circle passing through the vertices
A, B, and C of the parallelogram ABCD
intersects the lines AD and CD at points
M and N. The point M is at distances of
4, 3 and 2 from the vertices B, C, and D,
respectively. Find I MN I.
155. Given a triangle ABC in which
LBAC = n/6. The circle centred at A
with radius equal to the altitude dropped
on BC separates the triangle into two equal
areas. Find the greatest angle of the triangle
ABC.
156. In an isosceles triangle ABC LB =
120°. Find the common chord of two circles:
one is circumscribed about ABC, the other
passes through the centre of the inscribed
circle and the feet of the bisectors of the
angles A and C if I A C I = 1.
157. In a triangle ABC the side Be is
equal to a, the radius of the inscribed circle
is equal to r, Determine the radii of two
equal circles tangent to each other, one of
them touching the sides Be and BA, the
other-the sides Be and CA.
158. A trapezoid is inscribed in a circle

Sec. 1. Fundamental Facts.

35

of radius R. Straight lines passing through
the end points of one of the bases of the
trapezoid parallel to the lateral sides intersect at the centre of the circle. The lateral
side .can be observed from the centre at an
angle tX. Find the area of the trapezoid.
159. The hypotenuse of a right triangle
is equal to c. What are the limits of change
of the distance between the centre of the
inscribed circle and the point of intersection of the medians?
160. The sides of a parallelogram are
equal to a and b (a =1= b). What are the
limits of change of the cosine of the acute
angle between the diagonals?
161. Three straight lines are drawn
through a point M inside a triangle ABC
parallel to its sides. The segments of the
lines enclosed inside the triangle are equal
to one another. Find their length if the
sides of the triangle are a, b, and c.
162. Three equal circles are drawn inside
a triangle ABC each of which touches two
of its sides. The three circles have a common
point. Find their radii if the radii of the
circles inscribed in and circumscribed about
the triangle A Be are equal to rand R,
respectively.
163. In a triangle A·BC, a median AD
is drawn, LDAC + LABC = 90° Find
LBA"C if I AB I =1= I AC I·
164. Three circles of radii 1, 2, and 3
touch one another externally. Find the
3*

36


radius of the circle passing through the
points of tangency of these circles.
165. A square of unit area is inscribed
in an isosceles triangle, one of the sides
of the square lies on the base of the triangle.
Find the area of the triangle if the centres
of gravity of the triangle and square are
known to coincide.
166. In an equilateral triangle ABC,
the side is equal to a. Taken on the side Be
is a point D, and on the side AB a point E
such that 1 BD 1 = a/3, 1 AE 1 = I DE I.
Find 1 CE I·
167. Given a right triangle ABC. The
angle bisector CL (I CL I = a) and the
median
(I CM I = b) are drawn from
the vertex of the right angle C. Find the
area of the triangle ABC.
168. A circle is inscribed in a trapezoid.
Find the area of the trapezoid given the
length a of one of the bases and the line
segments band d into which one of the
lateral sides is divided by the point of
tangency (the segment b adjoins the base

cu

a).

169. The diagonals of a trapezoid are equal
to 3 and 5, and the line segment joining the
midpoints of the bases is equal to 2. Find
170. A circle of radius 1 is inscribed in a
triangle ABC for which cos B = 0.8. This
circle touches the midline of the triangle
ABC parallel to the side AC. Find AC.


37

171. Given a regular triangle ABC of
area S. Drawn parallel to its sides at equal
distances from them are three straight lines
intersecting inside the triangle to form a
triangle AIBICl whose area is Q. Find the
distance between the parallel sides of the
triangles ABC and AIBICl •
172. The sides AB and CD of a quadrilateral ABeD are mutually perpendicular;
they are the diameters of two equal circles
of radius r which touch each other. Find
the area of the quadrilateral ABCD if
I Be I : I AD I = k.
173. Two circles touching each other are
inscribed in an angle whose size is ct.
Determine the ratio of the radius of the
smaller circle to the radius of a third circle
touching both the circles and one of the
sides of the angle.
174. In a triangle ABC, circle intersecting the sides AC and BC at points M and
N, respectively, is constructed on the
Midline DE, parallel to AB, as on the
diameter. Find IMN I if IBC I = a, lAC I =
b, I AB 1= c.
175. The distance between the centres
of two circles is equal to a. Find the side of
a rhombus two opposite vertices of which
lie on one circle, and the other two on the
other if the radii of the circles are R
and r.
176. Find the area of the rhombus LiBeD
if the radii of the circles circumscribed


about the triangles ABC and ABD are R
and r, respectively.
177. Given an angle of size a with vertex
at A and a point B at distances a and b
from the sides of the angle. Find I AB I.
t 78. In a triangle ABC, the altitudes h a
and h b drawn from the vertices A and B,
respectively, and the length l of the bisector
of the angle C are given. Find Le.
right triangle. Another circle of the same
radius touches the legs of this triangle, one
of the vertices of the triangle being one of
the points of tangency. Find the ratio
of the area of the triangle to the area of
the common part of the two given circles.
180. Given ina trapezoidABCD: lAB 1=
I BC I = I CD I = a, I DA I = 2a. Taken
respectively on the straight lines AB
and AD are points E and F, other than the
vertices of the trapezoid, so that the point
of intersection of the altitudes of the
triangle CEF coincides with the point of
intersection of the diagonals of the trapezoid ABCD. Find the area of the triangle
CEF.

* •

*

18t. The altitude of a right triangle ABC
drawn to the hypotenuse AB is h, D being
its foot; M and N are the midpoints of the
line segments AD and DB, respectively.


39

Find the distance from the vertex C to the
point of intersection of the altitudes of the
triangle CMN.
182. Given an equilateral trapezoid with
bases AD and Be: I AB I = I CD I = a,
I AC I = I ED I = b, I Be I = c, M an
arbitrary point of the arc BC of the circle
circumscribed about ABeD. Find the ratio
'IBMI+IMCI
IAMI+IMDI
183. Each lateral side of an isosceles
triangle is equal to 1, the base being equal
to a. A circle is circumscribed about the
triangle. Find the chord intersecting the
lateral sides of the triangle and divided
by the points of intersection into three
equal segments.
184. MN is a diameter of a circle, IMNI=
1, A and B are points on the circle
situated on one side from M N, C is a point
on the other semicircle. Given: A is the
midpoint of semicircle, I MB I = 3/5, the
length of the line segment formed by the
intersection of the diameter MN with the
chords AC and Be is equal to a. What is
the greatest value of a?
185. ABCD is a convex quadrilateral.
M the midpoint of AB, N the midpoint of
CD. The areas of triangles ABN and CDM
are known to be equal, and the area of
their common part is 11k of the area of
each of them. Find the ratio of the sides
BC and AD.


40

186. Given an equilateral trapezoid ABCD
(AD II BC) whose acute angle at the larger
base is equal to 60°, the diagonal being
The point M is found at
equal to
distances 1 and 3 from the vertices A and
D, respectively. Find I Me I.
187. The bisector of each angle of a
triangle intersects the opposite side at a
point equidistant from the midpoints of
the two other sides of the triangle. Does it,
in fact, mean that the triangle is regular?
188. Given in a triangle are two sides:
a and b (a > b). Find the third side if it
is known that a
ha ~ b
h b , where h a
and h b are the altitudes dropped on these
sides (h a the altitude drawn to the side a).
189. Given a convex quadrilateral ABG'D
circumscribed about a circle of diameter 1.
Inside ABeD, there is a point M such that
2
2
2
I MA 1
1MB 12
1 Me 1
I MD 1 =
2. Find the area of ABCD.
tOO. Given in a quadrilateral ABeD:
I AB I = a, I BC I = b, I CD I = c, IDA 1=
d; a2 + c2 =1= b2 + tP, c =1= d, M is a
point on BD equidistant from A and C.
Find the ratio I BM 1 I MD I.
191. The smaller side of the rectangle
ABeD is equal to 1. Consider four concentric
circles centred at A and passing, respectively, through B, C, D, and the intersection
point of the diagonals of the rectangle
ABeD. There also exists a rectangle with

V3.

+

+

+

+

+


4t

vertices on the constructed circles ~one
vertice per circle). Prove that there IS a
square whose vertices lie on the constructed
circles. Find its side.
192. Given a triangle ABC. The perpendiculars erected to AB and Be at their midpoints intersect the line AC at points M
and N such that I MN I = I AC I. The
perpendiculars erected to AB and AC at
their midpoints intersect BC at points K
and L such that 1 KL I = ~ 1 Be I. Find
the smallest angle of the triangle ABC.
193. A point M is taken on the side AB
of a triangle ABC such that the straight
line joining the centre of the circle circumscribed about the triangle ABC to the
median point of the triangle BCM is perpendicular to cu. Find the ratio I BM I
I BA I if I BC I I BA I = k.
194. In an inscribed quadrilateral ABeD
where I AB I = I Be I, K is the intersection point of the diagonals. Find tAB I
if I BK I = b, I KD I = d.
195. Give the geometrical interpretations
of equation (1) and systems (2), (3), and
(4). Solve equation (1) and systems (2)
and (3). In system (4) find x + y + z:
(1) V x2+a2 - a x V 3

+ -V y2 + b2 -

by

Y3 .


42

+ V X 2+ y2_

XY

V3

=Va2 + b2 (a>O, b>O).
(2) { x = V Z2_ a2 + Vy2_a 2,
y == V x 2 - b2 + V z2- b2 ,
Z = V y2_ c2 + V x 2- c2 .
(3) x 2+ y2= (a-x2) +b2 = d2 + (b- y2.)
(4) { x 2 + xy + y2= a2,
y2 + yz + Z2 = b2,
Z2 + zx
x2 = a2 + b2 •

+

196. The side of a square is equal to a
and the products of the distances from the
opposite vertices to a line l are equal to
each other. Find the distance from the centre
of the square to the line l if it is known that
neither of the sides of the square is parallel
to l.
197. One of the sides in a triangle ABC
is twice the length of the other and LB =
2 LC. Find the angles of the triangle.
198. A circle touches the sides AB and
AC of an isosceles triangle ABC. Let M
be the point of tangency with the side AB
and N the point of intersection of the circle
and the base BC. Find I AN I if I AM I =
a, I BM I = b.
199. Given a parallelogram ABCD in
which I AB I = k I BC I, K and L are
points on the line CD (K on the side CD),


43

and M is a point on BC, AD being the
bisector of the angle KAL, AM the bisector
of the angle KAB, ) BM I = a, I DL I =
b. Find I AL I.
200. Given a parallelogram ABCD. A
straight line passing through the vertex C
Intersects the lines AB and AD at points
l( and L, respectively. The areas of the
,triangles KBC and CDL are equal to p
~nd q, respectively. Find the area of the
parallelogram ABeD.
201. Given a circle of radius R and two
points A and B on it such that I AB I = a.
Two circles of radii x and y touch the given
~$rcle at points A and B. Find: (a) the
length of the common external tangent to
the last circles if both of them touch the
~,:en circle in the same way (either interil,lly or externally); (b) the length of the
~JPmon internal tangent if the circle of
r~~ius x touches the given circle externally,
while the circle of radius y touches the
given circle internally.
.:. f202. Given in a triangle ABC: I AB I =
f2~, I BC I = 13, I CA I = 15. Taken on
~ side AC is a point M such that the
ii.' of the circles inscribed in the triangles
. AI and ReM are equal. Find the ratio

".'..
!
i"

,!1M I : I Me ).

.,. 203. The radii of the circles inscribed

and circumscribed about a triangle are
iiual to rand R, respectively. Find the
.'tea of the triangle if the circle passing

44


through the centres of the inscribed and
circumscribed circles and the intersection
point of the altitudes of the triangle is
known to pass at least through one of the
vertices of the triangle.
204. Given a rectangle AI1CD where
I AB I = 2a, I BC I = a
On the side
AR, as on diameter, a semicircle is constructed externally. Let M be an arbitrary
point on the semicircle, the line MD intersect AB at N, and the line Me at L. Find
2
2
1 AL 1 + I BN 1 (Fermat's" problem).
205. Circles of radii Rand r touch each
other internally. Find the side of the
regular triangle, one vertex of which coincides with the point of tangency, and the
other two, lying on the given circles.
206. Two circles of radii Rand r (R > r)
touch each other externally at a point A.
Through a point B taken on the larger
circle a straight line is drawn touching the
smaller circle at C. Find I Be I if I AB I =
a.
207. In a parallelogram ABeD there are
three pairwise tangent circles" *; one 0 f
them also touches the sides AB and Be, the
second the sides AB and AD, and the third
the sides Be and AD. Find the radius of
the third circle if the distance between the

V2.

• Fermat, Pierre de (t60t-1665), a French
amateur mathematician.
•• Any two of them have a point of tangency.


45

points of tangency on the side AB is equal
to a.
208. The diagonals of the quadrilateral
ABeD intersect at a point M, the angle
between them equalling Ct. Let 01' 02' 03'
0« denote the centres of the circles circumscribed about the triangles ARM, HeM,
CDM, DAM, respectively. Determine the
ratio of the areas of the quadrilaterals
ABeD and 1 2 3
209. In a parallelogram whose area is S,
the bisectors of its interior angles are drawn
to intersect one another. The area of the
quadrilateral thus obtained is equal to Q.
Find the ratio of the sides of the parallelogram.
2tO. In a triangle ABC, a point M is
taken on the side A C and a point N on the
ade BC. The line segments AN and BM
iBtersect at a point O. Find the area of the
.~angle CMN if the areas of the triangles
{)MA, DAB, and OBM are equal to 8 1 , S2'
ahd S3' respectively.
211. The median point of a right triangle
lies on the circle inscribed in this triangle.
B'i;nd the acute angles of the triangle.
212. The circle inscribed in a triangle
kBC divides the median BM into three
equal parts. Find the ratio I BC I I CA I
-lAB I213. In a triangle ABC, the midperpendic·alar to the side AB intersects the line AC
at M, and the midperpendicular to the side

°°°°..

46


AC intersects the line AB at N. It is known
that I MN I = I BC I and the line MN
is perpendicular to the line BC. Determine
the angles of the triangle ABC.
214. The area of a trapezoid with bases
AD and BC is S, I AD I : I BC I = 3;
situated on the straight line intersecting the
extension of the base AD beyond the point
D there is a line segment EF such that
AE II DF, BE II CF and I AE I : I DF I =
I CF I I BE I = 2. Determine the area of
the triangle EFD.
215. In a triangle ABC the side BC is
equal to a, and the radius of the inscribed
circle is r. Find the area of the triangle if
the inscribed circle touches the circle constructed on BC as diameter.
216. Given an equilateral triangle ABC
with side a, BD being its altitude. A second
equilateral triangle BDCl is constructed
on BD, and a third equilateral triangle
BDIC t is constructed on the altitude BD!
of this triangle. Find the radius of the circle
circumscribed about the triangle CClC 2 •
Prove that its centre is found on one of the
sides of the triangle ABC (C 2 is situated
outside the triangle ABC).
217. The sides of a parallelogram are
equal to a and b (a =1= b). Straight lines are
drawn through the vertices of the obtuse
angles of this parallelogram perpendicular
to its sides. When intersecting, these lines
form a parallelog~am similar to the given


47

one. Find the cosine of the acute angle of
the given parallelogram.
218. Two angle bisectors KN and LP
intersecting at a point Q are drawn in a
triangle KLM. The line segment PN has
a length of 1, and the vertex M lies on the
eircle passing through the points N, P,
and Q. Find the sides and angles of the
triangle PNQ.
. 219. The centre of a circle of radius r
touching the .sides AB, AD, and Be is
located on the diagonal AC of a convex
fluadrilateral ABeD. The centre of a circle
of' the same radius r touching the sides
Be, CD, and AD is found on the diagonal
BD. Find the area of the quadrilateral
~BCD if the indicated circles touch each
other externally.
220. The radius of the circle circumscribed
.bout an acute-angled triangle ABC is equal
lo t. The centre of the circle passing through
the vertices A, C, and the intersection point
"of the altitudes of the triangle ABC is known
:~ lie on this circle. Find I AC I.
°t 221. Given a triangle ABC in which
"ints M, N, and P are taken: M and N
~ the sides AC and BC, respectively, P
;.,. the line segment M N such that I AM I
j~.·MC I = I CN I : I NB I = I MP I : I P N I·
lind the area of the triangle ABC if the
.-eas of the triangles AMP and BNP are T
~d. Q, respectively.
~'~ 222. Given a circle of radius R and a
i

48


point A at a distance a from its centre
(a > R). Let K denote the point of the
circle nearest to the point A. A secant line
passing through A intersects the circle at
points M and N. Find I MN I if the area
of the triangle KMN is S.
223. In an isosceles triangle ABC
(I AB I = ,BC I), a perpendicular to AE
is drawn through the end point E of the
angle bisector AE to intersect the extension
of the side AC at a point F (C lies bet-veen
A and F). It is known that I AC I = 2m,
I FC I = m/4. Find the area of the triangle
ABC.

224. Two congruent regular triangles
ABC and CDE with side 1 are arranged
on a plane so that they have only one common point C, and the angle BCD is less than
n/3. K denotes the midpoint of the side
AC, L the midpoint of GE, and M the midpoint of RD. The area of the triangle KLM
is equal to V3/5. Find I BD I.
225. From a point K situated outside a
circle with centre 0, two tangents KM
and KN (M and N points of tangency) are
drawn. A point C (I Me I < I CN I) is
taken on the chord MN. Drawn through the
point C perpendicular to the line segment
OC is a straight line intersecting the line
segment N K at B. The radius of the circle
is known to be equal to R, LMKN = a,
1 MC I = b. Find I CB I·


49

226. A pentagon ABCDE is inscribed in
a circle. The points M, Q, N, and P are the
feet of the perpendiculars dropped from the
vertex E of the sides AB, Be, CD (or their
extensions), and the diagonal AD, respectively. It is known that I EP I = d, and the
ratio of the areas of the triangles MQE and
PNE is k. Find I EM f.
227. Given right trapezoid. A straight
line, parallel to the bases of the trapezoid
separates the latter into two trapezoids
such that a circle can be inscribed in each
of them. Determine the bases of the original
trapezoid if its lateral sides are equal to c
and d (d > c).
228. Points P and Q are chosen on the
lateral sides KL and M N of an equilateral
trapezoid KLMN, respectively, such that
the line segment PQ is parallel to the bases
of the trapezoid. A circle can be inscribed
in each of the trapezoids KPQN and
PLMQ, the radii of these circles being
equal to Rand r, respectively. Determine
the bases I LM I and I KN I·
the angle A intersects the side BC at a
point D. It is known that lAB I-IBD 1=
a, I AC I
I CD I = b. Find I AD I·
230. Using the result of the preceding
problem, prove that the square of the
bisector -of the triangle is equal to the product of the sides enclosing this bisector
minus the product of the line segments of

a

+

4-01557

50


the third side into which the latter is divided
by the bisector.
231. Given a circle of diameter A B. A second circle centred at A intersects the first
circle at points C and D and its diameter at
E. A point M distinct from the points C
and E is taken on the arc CE that does not
include the point D. The ray BM intersects
the first circle at a point N. It is known that
I CN I = a, I DN I = b. Find I MN I·
232. In a triangle ABC, the angle B is
'Jt/4, the angle C is :1/6. Constructed on the
medians BN and CN as diameters are circles intersecting each other at points P
and Q. The chord PQ intersects the side
BC at a point D. Find the ratio I BD I
IDC I.
233. Let AB denote the diameter of a
circle, 0 its centre, AB = 2R, C a point
on the circle, M a point on the chord AC.
From the point M, a perpendicular MN
is dropped on AB and another one is erected to AC intersecting the circle at L (the
line segment CL intersects AB). Find the
distance between the midpoints of AO
and CL if I AN I = a.
triangle ABC. A tangent to the circle
passing through the point B intersects the
line AC at M. Find the ratio I AM I
I Me I if I AB I : I BC I = k.
235. Points A, B, C, and D are situated
in consecutive order on a straight line,


5t

where I AC I = a, I AB I, I AD I = P I AB I·
An arbitrary circle is described through A
and B, CM and DN being two tangents to
this circle (M and N are points on the circle
lying on opposite sides of the line AB).
In what ratio is the line segment AB divided
by the line M N?
236. In a circumscribed quadrilateral
ABeD, each line segment from A to the
points of tangency is equal to a, and each
line segment from C to the points of tangency is b. What is the ratio in which the
diagonal AC is divided by the diagonal
BD?
237. A point K lies on the base AD of the
trapezoid ABeD. such that I AK I =
A I AD I. Find the ratio I AM I : I MD I,
where M is the point of intersection of the
base AD and the line passing through the
intersection points of the lines AB and CD
and the lines B K and A C.
Setting A = tin (n = 1, 2, 3,
.], divide a given line segment into n equal parts
using a straight edge only given a straight
line parallel to this segment.
238. In a right triangle ABC with the
hypotenuse AB equal to c, a circle is constructed on the altitude CD as diameter.
Two tangents to this circle passing through
the points A and B touch the circle at points
M and N, respectively, and, when extended,
intersect at a point K. Find I M K I.
239. Taken on the sides AB, Be and CA
4*

52


of a triangle ABC are points Cl , Al and B I
such
that
I ACt I : I CIB I = I BAt I :
I Al CI = I CB t I: I BtA I = k. Taken on
the sides AtB I , BIC I , and CIA I are points
C 2 , A 2 , and B 2 , such that I A IC 2 I
I GtB I , = I B IA 2 I I AtC I I = I
I
I B 2A I I = 11k. Prove that the triangle
A tB 2C 2 is similar to the triangle ABC and
find the ratio of similitude.
240. Given in a triangle ABC are the
radii of the circumscribed (R) and inscribed
(r) circles. Let At, B t , Cl denote the points
of intersection of the angle bisectors of the
triangle ABC and the circumscribed circle.
Find the ratio of the areas of the triangles
ABC and AIBICt .
241. There are two triangles with correspondingly parallel sides and areas 8 1 and 8 2 ,
one of them being inscribed in a triangle
ABC, the other circumscribed about this
t-riangle. Find the area of the triangle ABC.
242. Determine the angle A of the triangle
ABC if the bisector of this angle is perpendicular to the straight line passing
through the intersection point of the alti..
tudes of this triangle and the centre of the
circumscribed circle.
243. Find the angles of a triangle if the
distance between the centre of the circumscribed circle and the intersection point
of the altitudes is one-half the length of the
largest side and equals the smallest side.
244. Given a triangle ABC. A point D

c.s,


53

is taken on the ray BA such that I BD I =
I BA I + I AC I· Let K and M denote two
points on the rays BA and Be, respectively,
such that the area of the triangle BDM
is equal to the area of the triangle BCK.
Find LBKM if LBAC == Ct.
245. In a trapezoid ABeD, the lateral
side AB is perpendicular to AD and BC,
and I AB I = VI AD 1·1 BC I. LetE denote
the point of intersection of the nonparallel
sides of the trapezoid, 0 the intersection
point of the diagonals and M the midpoint
of AB. Find LEOM.
246. Two points A and B and two straight
lines intersecting at 0 are given in a plane.
Let us denote the feet of the perpendiculars
dropped from the point A on the given lines
by M and N, and the feet of the perpendiculars dropped from B by K and L, respectively. Find the angle between the lines MN
and KL if LAOB = Ct ~ 90°.
247. Two circles touch each other internally at a point A. A radius OB touching
the smaller circle at C is drawn from the
centre 0 of the larger circle. Find the angle
BAC.
248. Taken inside a square ABCD is a
point Msuch thatLMAB = 60°, LMCD =
15°. Find LMBC.
249. Given in a triangle ABC are two
angles: LA = 45° and LB = 15° Taken
on the extension of the side A C beyond the

54


point C is a point M such that I CM I =
2 I AC I. Find LAMB.
250. In a triangle ABC, LB = 60° and
the bisector of the angle A intersects BC
at M. A point K is taken on the side AC
such that LAMK = 30°. Find LOKe,
where 0 is the centre of the circle circumscribed about the triangle AMC.
251. Given a triangle ABC in which
1AB I = I AC I, LA = 80°. (a) A point
M is taken inside the triangle such that
LMBC = 30°, LMCB = 10°. Find LAMC.
(b) A point P is taken outside the triangle
such that LPBC = LPCA = 30°, and
the line segment BP intersects the side
AC. Find LPAC.
252. In a triangle ABC, LB = 100°,
LC = 65°; a point M is taken on AB such
that LMCB = 55°, and a point N is taken
on AC such that LNBC = BO°. Find

LNMC.
253. In a triangle ABC, I AB I = I BC I,
LB = 20°. A point M is taken on the side
AB such that LMCA = 60°, and a point N
on the side CB such that LNA C = 50°.
Find LNMC.
254. In a triangle ABC, LB = 70°,
LC = 50°. A point M is taken on the side
AB such that LMCB = 40°, and a point N
on the side AC such that LNBC = 50°.
Find LNMC.
255. Let M and N denote the points of
tangency of the inscribed circle with the


55

sides Be and BA of a triangle ABC, K the
intersection point of the bisector of the
angle A and the line MN. Prove that
LAKC = 90°.
256. Let P and Q be points of the circle
circumscribed about a triangle ABC such
2
2
that I PA 1 = I PB 1·1 PC I, I QA 1 =
I QR I· I QC I (one of the points is on the
arc AB, the other on the arc A C). Find
LPAB - LQAC if the difference between
the angles Band C of the triangle ABC
is (7,.
257. Two fixed points A and B are taken
on a given circle and ........AB = (7,. An arbitrary circle passes through the points A and
B. An arbitrary line 1 is also drawn through
the point A and intersects the circles at
points C and D different from B (the point
C is on the given circle). The tangents to
the circles at the points C and D (C and D
the points of tangency) intersect at M;
N is a point on the line 1 such that I eN I =
I AD I, I DN I = I CA I· What are the
values the LCMN can assume?
258. Prove that if one angle of a triangle
is equal to 120°, then the triangle formed
by the feet of i ts angle bisectors is rightangled.
259. Given in a quadrilateral ABCD:
LDAB = 150°, LDAC + LARD = 120°,
LDBC - LARD = 60°. Find LBDe.

56


• **
260. Given in a triangle ABC: I AB I =
1. I AC I = 2. Find I BC I if the bisectors
of the exterior angles A and C are known
to be congruent (i.e., the line segment of
the bisectors from the vertex to the intersection point with the straight line including the side of the triangle opposite to the
angle).
261. A point D is taken on the side CB
of a triangle ABC such that I CD I =
ex lAC [. The radius of the circle circumscribed about the triangle ABC is R. Find
the distance between the centres of the
ABC and ADB.
262. A circle is circumscribed about 8
right triangle ABC (LC = 90°). Let CD
denote the altitude of the triangle. A circle
centred at D passes through the midpoint of
the arc AB and intersects AB at M. Find
J cu I if I AB I = c.
263. Find the perimeter of the triangle
ABC if I BC I = a and the segment of the
straight line tangent to the inscribed circle
and parallel to Be which is enclosed inside
the triangle is b.
264. Three straight lines parallel to the
sides of a triangle and tangent to the inscribed circle are drawn. These lines cut
off three triangles from the given one. The
radii of the circles circumscribed about them

57

are equal to R 1 , R t , and R a• Find the radius
of the circle circumscribed about the given
triangle.
265. Chords AB and AC are drawn in a
circle of radius R. A point M is taken on
AB or on its extension beyond the point B,
the distance from M to the line A C being
equal to I AC I. Analogously a point N
is taken on AC or on its extension beyond
the point C, the distance from N to) the
line AB being equal to I AB I. Find MN.
266. Given a circle of radius R centred
at O. Two other circles touch the given
circle internally and intersect at points A
and B. Find the sum of the radii of these
two circles if LOAB = 90°.
267. Two mutually perpendicular intersecting chords are drawn in a circle of radius R. Find (a) the sum of the squares of
the four segments of these chords into
which they are divided by the point of intersection; (b) the sum of the squares of the
chords if the distance from the centre of
the circle to the point of their intersection
is equal to a.
268. Given two concentric circles of radii
rand R (r < R). A straight line is drawn
through a point P on the smaller circle to
intersect the larger circle at points Band
C. The perpendicular to Be at the point P
intersects the smaller circle at A. Find
2
2
2
I PA 1
I PB 1
I PC 1 •
269. In a semicircle, two intersecting

+

+

58


chords are drawn from the end points of the
diameter. Prove that the sum of the products of each chord segment that adjoins
the diameter by the entire chord is equal
to the square of the diameter.
270. Let a, b, C and d be the sides of an
inscribed quadrilateral (a be opposite to c),
h a , h b , he' and h d the distances from the centre of the circumscribed circle to' the corresponding sides. Prove that if the centre of
the circle is inside the quadrilateral, then
ah c + ch a = bhd + dh b •
271. Two opposite sides of a quadrilateral inscribed in a circle intersect at points
P and Q. Find I PQ I if the tangents to the
circle drawn from P and Q are equal to a
and b, respectively.
272. A quadrilateral is inscribed in a circle of radius R. Let P, Q, and M denote
the points of intersection of the diagonals'
of this quadrilateral with the extensions of
the opposite sides, respectively. Find the
sides of the triangle PQM if the distances
from P, Q, and M to the centre of the circle are a, b, and c, respectively.
273. A quadrilateral ABeD is circumscribed about a circle of radius r. The point of
tangency of the circle with the side AB
divides the latter into segments a and b,
and the point at which the circle touches
the side AD divides that side into segments
a and c. What are the limits of change of r?
274. A circle of radius r touches inter-

59

nally a circle of radius R, A being the point
of tangency. A straight line perpendicular
to the centre line intersects one of the circles at B, the other at C. Find the radius
of the circle circumscribed about the triangle ABC.
275. Two circles of radii Rand r intersect
each other, A being one of the points of intersection, BC a common tangent (B and
C points of tangency). Find the radius of
the circle circumscribed about the triangle ABC.
276. Given in a quadrilateral ABCD:
I AB I = a, I AD I = b; the sides BC, CD,
and AD touch a circle whose centre is in the
middle of AB. Find I BC I .
277. Given in, an inscribed quadrilateral ABCD: I AB I = a, I AD I = b (a >
b). Find I BC I if Be, CD, and AD
touch a circle whose centre lies on A B.

* • *
278. In a convex quadrilateral ABCD,
I AB I = I AD I. Inside the triangle
ABC, a point M is taken such that
LMBA = LADe, LMCA = LACD.
Find L MAC if L BAG = cx, L ADG L ACD = q>, I AM I < I AB I·
279. Two intersecting circles are inscribed
in the same angle, A being the vertex of
the angle, B one of the intersection points
of the circles, C the midpoint of the chord
whose end points are the points of tangency

60


of the first circle with the sides of the angle. Find the angle ABC if the common chord
can be observed from the centre of the
second circle at an angle a.
280. In an isosceles triangle ABC,
I AC I = 1 Be I, BD is an angle bisector, BDEF is a rectangle. Find L BAF
if L BAE = 1200
281. A circle centred at 0 is circumscribed
about a triangle ABC. A tangent to the circle at point C intersects the line bisecting
the angle B at a point K, the angle BKC being one-half the difference between the triple angle A and the angle C of the triangle.
The sum of the sides AC and AB is equal
to 2 + V3" and the sum of the distances
from the point 0 to the sides AC and AB
equals 2. Find the radius of the circle.
282. The points symmetric to the vertices of a triangle with respect to the opposite sides represent the vertices of the triangle with sides VB, VB, V14. Determine the
sides of the original triangle if their lengths
are different.
283. In a triangle ABC, the angle between
the median and altitude emanating from
the angle A is a, and the angle between the
median and altitude emanating from B
is ~. Find the angle between the median and
altitude emanating from the angle C.
284. The radius of the circle circumscribed about a triangle is R. The distance

Sec. 1. Fuudamental Facts

61

from the centre of the circle to the median
point of the triangle is d. Find the product
of the area of the given triangle and the
triangle formed by the lines passing
through its vertices perpendicular to the
medians emanating from those vertices.
285. The points AI' A 3 and As are situated
on one straight line, and the points A 2'
A., and A 8 on the other intersecting the
first line. Find the angles between these
lines if it is known that the sides of the
hexagon A IA 2A sA.Af)A. (possibly, a selfintersecting one) are equal to one another.
286. Two circles with centres 0 1 and O2
touch internally a circle of radius R centred at O. It is known that I 0 102 I = a.
A straight line touching the first two circles and intersecting the line segment 0 102
intersects their common external tangents
at points M and N and the larger circle at
points A and B. Find the ratio I AB I
I MNI if (8) the line segment 0 102 contains the point 0; (b) the circles with centres 0 1 and O2 touch each other",
287. The circle inscribed in a triangle
ABC touches the side A C at a point M and
the side BC at N; the bisector of the angle
A intersects the line MN at K, and the bisector of the angle B intersects the line M N
at L. Prove that the line segments MK,
N L, and KL can form a triangle. Find the
area of this triangle if the area of the
triangle ABC is S, and the angle C is a.

62


288. Taken on the sides AB and BC of a
square are two points' M and N such that
I BM I + I BN I = I AB I. Prove that
the lines DM and DN divide the diagonal
AC into three line segments which can form
a triangle, one angle of this triangle being
equal to 60°.
289. Given an isosceles triangle ABC,
I AB I = I BC I , AD being an angle bisector. '(he perpendicular erected to AD
at D intersects the extension of the side
AC at a point E; the feet of the perpendiculars dropped from Band D on AC are points
M and N, respectively. Find I MN I if
IAE I = a.
290. Two rays emanate from a point A
at an angle Cl. Two points Band B I are
taken on one ray and two points C and CI
on the other. Find the common chord of the
ABC and ABICI if I AB I - I AC I =
I ARI I - I AC 1 I = a.
291. Let 0 be the centre of a circle, C
a point on this circle, M the midpoint of
OC. Points A and B lie on the circle on the
same side of the line OC so that L AMO =
L BMC. Find
I AB I if I AM I IBM I = a.
292. Let A, B, and C be three points lying on the same line. Constructed on AR,
BC, and AC as diameters are three semicircles located on the same side of the line.
The centre of a circle touching the three


63

semicircles is found at a distance d from the
line AC. Find the radius of this circle.
293. A chord AB is drawn in a circle of
radius R. Let M denote an arbitrary point
of the circle. A line segment MN ( I MN I =
R) is laid off on the ray M A and on the
ray MB a line segment M K equal to the
distance from M to the intersection point
of the altitudes of the triangle MAB. Find
INK I if the smaller of the arcs subtended by AB is equal to 2a.
294. The altitude dropped from the right
angle of a triangle on the hypotenuse separates the triangle into two triangles in each
of which a circle is inscribed. Determine the
angles and the area of the triangle formed by
the legs of the original triangle and the line
passing through the centres of the circles
if the altitude of the original triangle is h.
295. The altitude of a fight triangle
drawn to the hypotenuse is equal to h.
Prove that the vertices of the acute angles
of the triangle and the projections of the
foot of the altitude on the legs all lie on the
same circle. Determine the length of the
chord cut by this circle on the line containing the. altitude and the segments of the
chord into which it is divided by the hypotenuse.
296. A circle of radius R touches a
line l at a point .A, AB is a diameter
of this circle, Be is an arbitrary chord.
Let D denote the foot of the perpendicular


dropped from C on AB. A point E lies on the
extension of CD beyond the point D, and
I ED I = I BC I The tangents to the circle, passing through E, intersect the line
l at points K and N. Find I KN I
297. Given in a convex quadrilateral
ABeD: I AB I = a, I AD I = b, I Be I ~
p - a, I DC I = p - b. Let 0 be the
point of intersection of the diagonals. Let
us denote the angle BA C by a. What does
I AO I tend to as a ~ O?

Section 2

Selected Problems and
Theorems of Plane Geometry

Carnot's Theorem

t. Given points A and B. Prove that the
locus of points M such that I AM I 2 I MB I 2 = k (where k is a given number)
is a straight line perpendicular to AB.
2. Let the distances from a point M to
the vertices A, B, and C of a triangle ABC
be a, b, and c, respectively. Prove that there
is no d =1= 0 and no point on the plane
for which the distances to the vertices in the
same order can be expressed by the numbers
Va2 d, b2 d, c2 d.
3. Prove that for the perpendiculars dropped from the points At, B 1 , and Cion the
sides Be, CA, and AB of a triangle ABC
to intersect at the same point, it is necessary and sufficient that

+

2
I AlB 1
2
1 BtC 1

V + V +

-

2
I BC I 1
I CAl 1 2

+ I CIA

2
1_

I ABI

2
1

+

= 0 (Carnot's theorem).

4. Prove that if the perpendiculars drop...
ped from the points AI' B t , and Ct on the
sides BC, CA, and AB of the triangle ABC,
respectively, intersect at the same point,
then the perpendiculars dropped from the
5 -01557

66


points A, B, and C on the lines BICI, CIA I,
and AIB I also intersect at one point.
5. Given a quadrilateral ABeD. Let AI,
B l , and C1 denote the intersection points
of the altitudes of the triangles BCD,
ACD, and ABD. Prove that the perpendiculars dropped from A, B, and C on the
lines BICI , CIAI , and AtB I , respectively,
intersect at the same point.
6. Given points A and B. -Prove that the
locus of points M such that k I AM 1 2 +
l I BM I 2 = d (k, l, d given numbers,
k + l =1= 0) is either a circle with centre on
the line AB or a point or. an empty set.
7. Let At, A 2 , •• , An be fixed points
and k l , k 2 , . ., k n be given numbers. Then
the locus of points M such that the sum
2
k l I AIM 1 2 + k 2 I A 2M 1
+ · · +
k n I A n M I 2 is constant is: (a) a circle, a
point, or an empty set if k l + k 2 + .
+
k n =1= 0; (b) a straight line, an empty set,
+
or the entire plane if k l + k 2 +
k; -= o.
8. Given a circle and a point A outside
the circle. Let a circle passing through A
touch the given circle at an arbitrary point
B, and the tangents to the second circle
which are drawn through the points A and
B intersect at a point M. Find the locus of
points M.
9. Given points A and .B. Find the locus
of points M such that I AM I I MB I =
k =1= 1.

Sec. 2. Selected Problems

67

10. Points A, B, and C lie on a straight
line (B between A and C). Let us take an arbitrary circle centred at B and denote by M
the intersection point of the tangents drawn
from A and C to that circle. Find the locus
of points M such that the points of tangency
of straight lines AM and CM with the circle belong to the open intervals AM and GM.
1I , Given two circles. Find the locus of
points M such that the ratio of the lengths
of the tangents drawn from M to the given
circles is a constant k.
12. Let a straight line intersect one circle at points A and B and the other at
points C and D. Prove that the intersection
points of the tangents to the first circle
which are drawn at points A and B and the
tangents drawn to the second circle at
points C and D (under consideration are the
intersection points of tangents to distinct
circles) lie on a circle whose centre is found
on the straight line passing through the centres of the given circles.
13. Let us take three circles each of which
touch one side of a triangle and the extensions of two other sides. Prove that the perpendiculars erected to the sides of the triangle at the points of tangency of these circles intersect at the same point.
14. Given a triangle ABC. Consider all
possible pairs of points M 1 and M 2 such that
I AMt I : I BMt I : I CM t I = I AMI I :
I BM 2 I I CM 2 I· Prove that the lines
5*


MIMI pass through the same fixed point
in the plane.
t5. The distances from a point M to the
vertices A, B, and C of a triangle are equal
to 1, 2, and 3, respectively, and from a
point M 1 to the same vertices to 3, 15, 5,
respectively. Prove that the straight line
M M l passes through the centre of the circle circumscribed about the triangle ABC.
16. Let AI' B l , Gl denote the feet of the
perpendiculars dropped from the vertices
A, B, and C of a triangle ABC on the line l,
Prove that the perpendiculars dropped from
AI' B l , and Cion Be, CA, and AB, respectively, intersect at the same point.
17. Given a quadrilateral triangle ABC
and an arbitrary point D. Let AI' B l , and
C1 denote the centres of the circles inscribed
in the triangles BCD, CAD, and ABD,
respectively. Prove that the perpendiculars
dropped from the vertices A, B, and C
on BlG l, CIA t , and AIBl , respectively, intersect at the same point.
18. Given three pairwise intersecting circles. Prove that the three common chords of
these circles pass through the same- point.
19. Points M and N are taken on lines AB
and AC, respectively. Prove that the common chord of two circles with diameters
eM and BN passes through the intersection
point of the altitudes of the triangle ABC.
20. A circle and a point N are given in a

V


69

plane. Let AB be an arbitrary chord of the
circle. Let M denote the point of intersection of the line AB and the tangent at the
point N to the circle circumscribed about
the triangle ABN. Find the locus of points
M.
21. A point A is taken inside a circle.
Find the locus of the points of intersection
of the tangents to the circle at the end points
of all possible chords passing through the
point A.
22. Given numbers a, p, y, and k. Let
x, y, I denote the distances from a point M
taken inside a triangle to its sides. Prove
that the locus of points M such that ax +
py
= k is either an empty set or
a line segment or coincides with the set of
all points of the triangle.
23. Find the locus of points M situated
inside a given triangle and such that the
distances from M to the sides of the given
triangle can serve as sides of a certain triangle.
24. Let AI' B I , and C1 be- the midpoints
of the sides BC, CA, and AB of a triangle
ABC, respectively. Points A 2 , B 2 , and C2
are taken on the perpendiculars dropped
from a point M on the sides BC, CA, and
AR, respectively. Prove that the perpendiculars dropped from At, B t , and Cion
the lines B tC 2 , C2A'}" and A tB 2 , respectively, intersect at the same point.
25. Given a straight line l and three

+ "1

70


lines It, l2' and i, perpendicular to l. Let
A, B, and C denote three fixed points on
the line l, At an arbitrary point on ll' B 1
an arbitrary point on ll' C1 an arbitrary
point on ls. Prove that if at a certain arrangement of the points At, B I , and C1 the perpendiculars dropped from A, B, and C
on the lines BIG I , CIA I , and AtBI , respectively, intersect at one and the same point,
then these perpendiculars meet in the same
point at any arrangement of At, B t , G1 •
26. Let AA I , BBI , GCI be the altitudes of
a triangle ABC, AI' B i , and C2 be the projections of A, B, and C on BIGI, CIA I,
and AIB I , respectively. Prove that the perpendiculars dropped from A 2 , B 2 , and C"
on BC, CA~, and AR, respectively, intersect at the same point.
Ceva's* and Menelaus'** Theorems.
Affine Problems

2:1. Prove that the area of a triangle whose
sides are equal to the medians of a given
triangle amounts.. to 3/4 of the area of the
latter.
• Ceva, Giovanni (t647-t 734). An Italian
mathematician who gave static and geometric
proofs for concurrency of straight lines through
vertices of triangles.
•• Menelaus of Alexandria (first cent. A.D.).
A geometer who wrote several books on plane and
spherical triangles, and circles.


71

straight line parallel to BC intersects AB
and CD at points E and F, respectively,
and a straight line parallel to AB intersects
BC and DA at points G and H, respectively.
Prove that the lines EH, GF, and BD either
intersect at the same point or are parallel.
29. Given four fixed points on a straight
line l : A, B, C, and D. Two parallel lines
are drawn arbitrarily through the points
A and B, another two through C and D.
The lines thus drawn form a parallelogram.
Prove that the diagonals of that parallelogram intersect l at two fixed points.
30. Given a quadrilateral ABeD. Let
o be the point of intersection of the diagonals AC and BD, M a point on AC such that
I AM ( = I OC (, N a point on BD such
that I BN I = I OD I , K and L the midpoints of AC and BD. Prove that the lines
ML, N K, and the line joining the median
points of the triangles ABC and ACD intersect at the same point.
31. Taken on the side Be of a triangle
ABC are points Al and A 2 which are symmetric with respect to the midpoint of BC.
In similar fashion taken on the side AC are
points BI and B 2 , and on the side AB
points C1 and C2 • Prove that the triangles
AIBICl and A 2B 2 C2 are equivalent, and
the centres of gravity of the triangles
AlBIC!, A 2B 2C 2 , and ABC are collinear.
32. Drawn through the intersection point

72


M of medians of a triangle ABC is a
straight line intersecting the sides AB and
AC at points K and. L, respectively, and
the extension of the side BC at a point P (C
lying between P and B). Prove that I~KI =
1

1

IMLI + IMPI ·
.
of the diagonals of a quadrilateral ABCD
is a straight line intersecting AB at a point
M and CD at a point N. Drawn through
the points M and N are lines parallel to
CD and AR, respectively, intersecting AC
and BD at points E and F. Prove that BE
is parallel to CF.
34. Given a quadrilateral ABCD. Taken
on the lines AC and BD are points K and
M, respectively, such that BK II AD and
AM II BC. Prove that KM II CD.
35. Let E be an arbitrary point taken on
the side AC of a triangle ABC. Drawn
through the vertex B of the triangle is an
arbitrary line l. The line passing through the
point E parallel to Be intersects the line
l at a point N, and the line parallel to AB
at a point M. Prove that AN is parallel to
CM.

• • •
36. Each of the sides of a convex quadrilateral is divided into (2n
1) equal parts.
The division points on the opposite sides

+


73

are joined correspondingly. Prove that the
area of the central quadrilateral amounts to
1/(2n + 1)2 of the area of the entire quadrilateral.
37. A straight line passing through the
midpoints of the diagonals AC and BC of a
quadrilateral ABeD intersects its sides AB
and DC at points M an N, respectively.
Prove that 8 D C M = SASH·.
38. In a parallelogram ABeD, the vertices A, B, C, and D are joined to the midpoints of the sides CD, AD, AB, and
Be, respectively. Prove that the area of
the quadrilateral formed by these line segments is 1/5 of the area of the parallelogram.
39. Prove that the area of the octagon
formed by the lines joining the vertices of a
parallelogram to the midpoints of the opposite sides is 1/6 of the area of the parallelogram.
40. Two parallelograms ACDE and
BCFG are constructed externally" on the
sides AC and BC of a triangle ABC. When
extended, DE and FD intersect at a point
H. Constructed on the side AB is a parallelogram ABML, whose sides AL and BM are
equal and parallel to HC. Prove that the
parallelogram ARML is equivalent to the
sum of the parallelograms constructed on
AC and Be.
• Here and elsewhere, such a notation symbolizes the area of the figure denoted by the subscript.

74


41. Two parallel lines intersecting the larger base are drawn through the end points
of the smaller base of a trapezoid. Those
lines and the diagonals of the trapezoid separate the trapezoid into seven triangles and
one pentagon. Prove that the sum of the
areas of the triangles adjoining the lateral
sides and the smaller base of the trapezoid
is equal to the area of the pentagon.
42. In a parallelogram ABCD, a point E
lies on the line AB, a point F on the line
AD (B on the line segment AE, D on AF),
K being the point of intersection of the
lines ED and FB. Prove that the quadrilaterals ABKD and CEKF are equivalent.

* * *
43. Consider an arbitrary triangle ABC.
Let AI' B 1 , and C1 be three points on the
lines BC, CA, and AB, respectively. Using
the following notation

R = I ACt I'. I BA I I . I CB 1 I
I CtB I

R* _ sin
-

I AIC I

LACC l

sin LelCB

•

I BlA I '

sin LBAA I
sin LA1AC

•

sin LCBB1
sin LBIBA '

prove that R = R*.
44. For the lines AA l t BB t , eel to meet
in the same point {or for all the three to be
parallel), it is necessary and sufficient that
R = 1 (see the preceding problem), and of
three points At., B., Ct the one or all the

75


three lie on the sides of the triangle ABC,
and not on their extensions (Ceva's theorem).
45. For the points AI' B I , C1 to lie on the
same straight line, it is necessary and sufficient that R = 1 (see Problem 43, Sec. 2),
and of three points AI' B t t CI no points or
two lie on the sides of the triangle ABC,
and not on their extensions (Menelaus'
theorem).

Remark. Instead of the ratio : ~~~: and
the other two, it is possible to consider
the ratios of directed line segments which
AC
are denoted by Cl~ and defined as foI-

I CtB I
ACt

I ACt I
ACt iti
I CtB I ' CtB IS POSI rve
---.
when the vectors ACt and CiB are in the
same direction and ~~~ negative if these
vectors
are
in opposite directions.

l ows.·

.:.=

~

(~~~ has sense only for points situated
on the same straight line.)

It is easily

seen that the ratio ~~; is positive if the
point C1 lies on the line segment AB and
the ratio is negative if C, is outside AB.
Accordingly, instead of R, we shall consider the product of the ratios of directed line
segments which is denoted by If. Further,
we introduce the notion of directed angles.

76


For instance, by 4ACCI we shall understand
the angle through which we have to rotate
CA about C anticlockwise to bring the ray
CA into coincidence with the ray eel.
Now, instead of R* we shall consider the
product of the ratios of the sines of directed
angles j •.
Now, we have to reformulate Problems
43, 44, and 45 of this Section- in the following way:
43*. Prove that j = j •.
44*. For the lines AAl t BB t , eel to meet
in the same point (or to be parallel), it is
that R = 1
necessary and sufficient

(Ceva's theorem).
•
45*. For the points AI' Bit C1 to be col-

linear, it is necessary and sufficient that

R=

- 1 (Menelaus' theorem).
46. Prove that if three straight lines, passing through the vertices of a triangle, meet
in the same point, then the lines symmetric to them with respect to the corresponding angle bisectors of the triangle also
intersect at one point or are parallel.
47. Let 0 denote an arbitrary point in a
plane, M and N the feet of the perpendiculars dropped from 0 on the bisectors of the
interior and exterior angle A of a triangle
ABC; P and Q are defined in a similar manner for the angle B; Rand T for the angle
C. Prove that the lines MN, PQ, and RT


77

intersect at the same point or are parallel.
48. Let 0 be the centre of the circle inscribed in a triangle ABC, A o, B o, Co
the points of tangency of this circle with the
sides BC, CA, AB, respectively. Taken on
the rays OA o,l OB o, OCo are points L, M,
K, respectively, equidistant from the point
O. (a) Prove that the lines AL, BM, and
CK meet in the same point. (b) Let AI'
Bit C, be the projections of A, B, C, respectively, on an arbitrary line l passing
through O. Prove that the lines AlL, B IM,
and CIK are concurrent (that is, intersect
at a common point).
49. For the diagonals AD, BE, and CF
of the hexagon ABCDEF inscribed in a circle to meet in the same point, it is necessaryand sufficient that the equality I AB I X
I CD I · I EF' = I Be I · I DE I I FA I
be fulfilled.
50. Prove that: (a) the bisectors of the exterior angles of a triangle intersect the extensions of its opposite sides at three points
lying on the same straight line; (b) the tangents drawn from the vertices of the triangle to the circle circumscribed about it intersect its opposite sides at three collinear
points.
51. A circle intersects the side AB of a
triangle ABC at points C1 and Cit the side
CA at points BI and B I , the side Be at
points.d, andA.e Prove that if the li~esAAlt

78


BB t , and CCI meet in the same point, then
the lines AA 2 , BB 2 , and CC2 also intersect
at the same point.
52. Taken on the sides AB, BC, and CA
of a triangle ABC are points CI , AI' and
8 1 • Let C2 be the intersection point of the
lines AB and AIBt , At the intersection point
of the lines BC and BtCt, B 2 the intersection point of the lines AC and AICI .
Prove that if the lines AA t , BB l , and
eCI meet in the same point, then the
points AI' B 2 , and C 2 lie on a straight
line.
53. A straight line intersects the sides AB,
BC, and the extension of the side AC of
a triangle ABC at points D, E, and F, re-_
spectively. Prove that the midpoints of the
line segments DC, AE, and BF lie on a
straight line (Gaussian· line).
54. Given a triangle ABC. Let us define
a point Al on the side BC in the following
way: Al is the midpoint of the side KL
of a regular pentagon M KLNP whose vertices K and L lie on BC, and the vertices M
and N on AB and AC, respectively. Defined in a similar way on the sides AB and
AC are points Ct and B t • Prove that the
lines AA 1 , BB l , and CCI intersect at the
same point.

• Gauss, Carl Friedrich (1777-1855). A German matbematician.


79

55. Given three pairwise'" nonintersecting circles. Let us denote by AI' At, A a
the three points of intersection of common
internal tangents -to any two of them and by
B l , B 2 , B 3 the corresponding points of intersection of the external tangents. Prove that
these points are situated on four lines, three
on each of them (AI' A 2 , B a; AI' B 2 , A 3 ;
s; A 2 , A 3 ; B l , B 2 , B 3 ) ·
56. Prove that if the straight lines passing through the vertices A, B, and C of a
triangle ABC parallel to the lines BtCl ,
CIAl , and AIBI meet in the same point,
then the straight lines passing through AI;
B I , and CI parallel to the lines BC, CA,
and AB also intersect at the same point (or
are parallel).
57. Given a triangle ABC, M being an arbitrary point in its plane. The bisectors of
two angles formed by the lines AM and BM
intersect the line AB at points CI and C2
(Cl lying on the line segment AB), determined similarly on BC and CA are points
AI' A 2 , and B l , B 2 , respectively. Prove
that the points AI' A 2 , B l , B i , CI , Ct are
situated on four lines, three on each of them.
58. Points AI' B l , CI are taken on the
sides BC, CA, and AB of a triangle ABC,
respectively, and points At, B t , C2 on the
sides BIC I , CIAI, AIBI of the triangle
AIBICI- The lines AA l , BB l , eCI meet in
• No two of which intersect.

80


the same point, and the lines AlAI' BlB a,
G1C., also intersect at one point. Prove that

the lines AA s' BB 2 cel' are either concur,
rent or parallel.
59. In 8 quadrilateral ABCD, P is the
intersection point of Be and AD, Q that of
CA andBD, andR thatofAB and CD. Prove
that the intersection points of BC and
QR, CA and RP, AB and PQ are collinear.
60. Given an angle with vertex O. Points
AI' As, As, A, are taken on one side of the
angle and points B l , B a, Bat B, on the other
side. The lines AlB I and A 2B a intersect at
a point N, and the lines AsB s and A,B,
at a point M. Prove that for the points
0, Nand M to be collinear, it is necessary
and sufficient that the following equality·
be fulfilled:
OBI
BaB,
OBi • DB, • B 1B1

OBI

=

OA I
OA2 AsA.
OAa • OA, · AlAI •

(See Remark to Problems 43-45).
61. Given a triangle ABC. Pairs of points
Al and AI' B 1 and B t , C1 and Ca are taken
on the sides BC, GA, and AB, respectively,
such that AA I , BB1 , and Gel meet in the
same point, and AA., BB., and GCI also
intersect at one point. Prove that: (8) the
points of intersection of the lines AIB l and
AB, BIGl and sc, CIA I and GA lie on a
straight line ll. Just in the same way, the
points A~1t B I ., and C2 determine a straight
line lit (b) the point A, the intersection point


81

of the lines II and l. and the intersection
point of the lines BIGI and BsG s lie on one
straight line; (c) the intersection points of
the lines Be and BIC I , CA and CIA., AB
and AIBI are collinear.
62. An arbitrary straight line intersects
the lines AB, BC, and CA at points K, M,
and L, respectively, and the lines AIBI ,
BIC I , and CIA I at points K I , MI , and Lt.
Prove that if the lines AtM, BtL, and CIK
meet in the same point, then the lines AMI'
BLI , and CKt are also concurrent.
63. Given a triangle ABC and a point D.
Points E, F, and G are situated on the lines
AD, /J:D, and CD, respectively, K is the intersection point of AF and BE, L the intersection point of BG and CF, M the intersection point of CE and AG, P, Q, and R
are the intersection points of DK and AB,
DL and BC, DM and AC. Prove that all
the six lines AL, EQ, BM, FR, CK, and
GP meet in the same point.
64. The points A and At are symmetric
with respect to a line l, as are the pairs B
and B t , C and CI , and N is an arbitrary
point on l. Prove that lines AN, BN, eN
intersect, respectively, the lines BtCI ,
CtA I , and AIBl" at three points lying on a
straigh t line.
65. Let At, As, Ar. be three points situated on one straight line, and As, A"A. on the
other. Prove that the three points at which
the pairs of lines 4142 and. A t A 6' A 2 A 8
6-01557

82


and A&A" AsA. and A,A I intersect lie on
a straight line.
Loci of Points
of two circles is a straight line intersecting
the circles for the second time at points A
and B. Find the locus of the midpoints of
AB.
67. Given a point A and a straight line I,
B being an arbitrary point on I. Find the
locus of points M such that ABM is a regular triangle.
68. Given a regular triangle ABC. Points
D and E are taken on the extensions of its
sides AB and A C beyond the points Band C,
respectively, such that I BD I I CE I =
I BC I 2. Find the locus of the points of
intersection of the lines DC and BE.
69. Given three points A, B, and C on a
straight line, and an arbitrary point D
in a plane not on the line. Straight lines parallel to AD and BD intersecting the lines
BD and AD at points P and Q are drawn
through the point C. Find the locus of the
feet M of perpendiculars dropped from C
on PQ, and find all the points D for which
M is a fixed point.
70. A point K is taken on the side AC
of a triangle ABC and point P on the median BD such that the area of the triangle
APK is equal to the area of the triangle


83

BPe. Find the locus of the intersection
points of the lines AP and BK.
71. Two rays forming a given angle a are
passing through a given point 0 inside a
given angle. Let one ray intersect one side
of the angle at a point A, and the other
ray the other side of the angle at a point B.
Find the locus of the feet of the perpendiculars dropped from 0 on the line AB.
72. Two mutually perpendicular diameters AC and BD are drawn in a circle. Let
P be an arbitrary point of the circle, and
let PA intersect BD at a point E. The
straight line passing through E parallel to
AC intersects the line PB at a point M.
Find the locus of points M.
73. Given an angle with vertex at A
and a point B. An arbitrary circle passing
through the points A and B intersects the
sides of the angle at points C and D (different from A). Find the locus of the centres
of mass of triangles ACD.
74. One vertex of a rectangle is found at a
given point, two other vertices, not belonging to the same side, lie on two given
mutually perpendicular straight lines. Find
the locus of fourth vertices of such rectangles.
75. Let A be one of the two intersection
points of two given circles; drawn through
the other point of intersection is an arbitrary line intersecting one circle at a point B
and the other at a point C, both points differ-

84


ent from common points of these circles.
Find the locus of: (a) the centres of the circles circumscribed about the triangle ABC;
(b) the centres of mass of the triangles ABC;
(c) the intersection points of the altitudes of
the triangle ABC.
76. Let Band C be two fixed points of 8
given circle and A a variable point of this
circle. Find the locus of the feet of the perpendiculars dropped from the midpoint of
AB on AC.
77. Find the' locus of the intersection
points of the diagonals of rectangles whose
sides (or their extensions) pass through four
given points in the plane.
78. Given two circles touching each other
internally at a point A. A tangent to the
smaller circle intersects the larger one at
points Band C. Find the locus of centres of
circles inscribed in triangles ABC.
79. Given two intersecting circles. Find
the locus of centres of rectangles with vertices lying on these circles.
80. An elastic ball whose dimensions may
be neglected is found inside a round billiard table at a point A different from the
centre. Indicate the locus of points A from
which this ball can be directed so that after
three successive boundary reflections, bypassing the centre of the billiard table, it
finds itself at the point A.
81. Through a point equidistant from two
given parallel lines a straight line is drawn


85

intersecting these lines at points M and N.
Find the locus of vertices P of equilateral
triangles M NP.
82. Given two points A and B and a
straight line I. Find the locus of the centres
of circles passing through A and B and intersecting the line l.
83. Given two points 0 and M. Determine: (8) the locus of points in the plane
which can serve as one of the vertices of a
triangle with the centre of the circumscribed
circle at the point O~and the centre of mass
at the point M; (b) the locus of points in
the plane which can serve 8S one of the
vertices of an obtuse triangle with the centre'of the circumscribed circle at the point 0
and the centre of mass at the point M.
M. An equilateral triangle is inscribed
in a circle. Find the locus of intersection
points of the altitudes of all possible triangles inscribed in the circle if two sides of
the triangles are parallel to those of the
given one.
85. Find the locus of the centres of all
possible rectangles circumscribed about a
given triangle. (A rectangle will be called
circumscribed if one of the vertices of the
triangle coincides with a vertex of the rectangle, and two others lie on two sides of
the rectangle not including this vertex.)
86. Given two squares whose sides are
respectively parallel. Determine the locus
of points M such that for any point P of the

86

Problems ,in Plane Geometry

first square there is a point Q of the second
one such that the triangle MPQ is equilateral. Let the side of the first square be a
and that of the second square is b. For what
relationship between a and b is the desired
locus not empty?
87. Inside a given triangle, find the locus
of points M 'for each of which and for any
point N on the boundary of the triangle there
is a point P, inside the triangle or on its
boundary, such that the area of the triangle
MNP is not less than 1/6 of the area of the
given triangle.
88. Given two points A and I. Find the
locus of poi.nts B such that there exists a
triangle ABC with the centre of the inscribed
circle at the point I, all of whose angles
are less than ex (60° < a, < 90°).
89. Points A, B, and C lie on the same
straight line (B is found between A and
C). Find the locus of points M such that
cot LAMB
cot LBMC ~ k,
90. Given two points A and Q. Find the
locus of points B such that there exists an
acute triangle ABC for which Q is the centre
of mass.
91. Given two points A and H. Find
the locus of points B such that there is a
triangle ABC for which H is the point of intersection of its altitudes, and each of
whose angles is greater than a, (ex < n/4).
92. Given two rays in a plane. Find the
locus of points in the plane equidistant

+


87

from these rays. (The distance from a point
to a ray is equal to the distance from
this point to the nearest point of the ray.)
93. Given an angle and a circle centred at
o inscribed in this angle. An arbitrary line
touches the circle and intersects the sides
of the angle at points M and N. Find the locus of the centres of circles circumscribed
about the triangle MON.
94. Given two circles and two points A
and B (one on eifher circle) equidistant
from the midpoint of the line segment joining their centres. Find the locus of the midpoints of line segments AB.
95. Given aline segment AB. Let us take
an arbitrary point M on AB and consider
two squares AMen and MBEF situated on
the same side of AB. We then circumscribe
circles about these squares and denote the
point of their intersection by N (N is different from M). Prove that: (8) AF and Be
intersect at N; (b) MN passes through a
fixed point in the plane. Find the locus of
the midpoints of line segments joining the
centres of the squares.
96. Given a circle and a point A. Let M
denote an arbitrary point on the circle.
Find the locus of points of intersection of
the midperpendicular to the line segment
AM and the tangent to the circle passing
through the point Af.
97. Two circles touch each other at a
point A. One line passing through A inter-

88


sects these circles for the second time at
points Band C, the other line-at points B 1
and C1 (B and B 1 lie on the same circle).
Find the locus of points of intersection of
the circles circumscribed about the triangles ABIC and ABCI •
98. Find the locus of the vertices of right
angles of all possible right isosceles triangles the end points of whose hypotenuses
lie on two given circles.
99. The sides of a given triangle serve as
diagonals of three parallelograms. The
sides of the parallelograms are parallel to
two straight lines land p. Prove that the
three diagonals of these parallelograms,
different from the sides of the triangle, intersect at a point M. Find the locus of
points M if land p are arbitrary and mutually
perpendicular.
tOO. Let Band C denote two fixed points
of a circle, A being an arbitrary point of the
circle. Let H be the intersection point of the
altitudes of the triangle ABC and M be the
projection of H on the bisector of the angle
BAC. Find the, locus of points M.
tot. Given a triangle ABC. Let D be an
arbitrary point on the line Be. Straight
lines passing through D parallel to AB and
AC intersect AC and AB at points E and
F, respectively. Find the locus of the centres of circles passing through the points
D, E, and F.
102. Given a regular triangle ABC.


89

Find the locus of points M inside the trianL MAB + L MBG +
gle such that
LMCA = n/2.
103. A point M is taken inside a triangle
such that there is a straight line l passing
through M and separating the triangle into
two parts so that in the symmetric mapping
with respect to lone part turns out to be inside, or at the boundary of, the other. Find
the locus of points AI.
Triangles. A Triangle and a Circle
104. From the vertex A of a triangle ABC,
perpendiculars AM and AN are dropped on
the bisectors of the exterior angles Band
C of the triangle. Prove that the line segment MN is equal to half the perimeter of
the triangle ABC.
105. An altitude BD is drawn in a triangle ABC, AN is perpendicular to AB, eM
is perpendicular to BC, and I AN I =
I DC I, I cu I = I AD I. Prove that
M and N are equidistant from the vertex

B.

106. Prove that for any right triangle the
radius of the circle which touches internally the circumscribed circle and the legs is
equal to the diameter of the inscribed circle.
107. Prove that if one of the sides of
a triangle lies on a fixed line in 8 plane and if
the point of intersection of the altitudes coin-

90


cides with the fixed point, then the circle
circumscribed about this triangle also
passes through the fixed point.
108. Given a triangle ABC. LetA I , Bland
C1 be the points of the circle circumscribed
about ABC and diametrically opposite
to the vertices A, B, and C, respectively.
Straight lines parallel to BC, CA, and AB
are drawn through AI' B I , and CI , respectively. Prove that the triangle formed by
these lines is homothetic to the triangle
ABC, with the ratio of similitude 2 and
centre at the intersection point of the altitudes of the triangle ABC.
109. Prove that the projections of the foot
of the altitude of a triangle on the sides
enclosing this altitude and on the two other
altitudes lie on one straight line.
110. In a triangle ABC, a point D
is taken on the side AB extended beyond
the point B such that I BD I = I CB I
In the same manner, taken on the extension
of the side CB beyond the point B is 8
point F such that I BF I = I AB I. Prove
that the points A, C, D, and F lie on the
same circle whose centre is found on the
circle circumscribed about the triangle
ABC.
111. Three equal circles pass through a
point H. Prove that H is the intersection
.point of the altitudes of the triangle whose
vertices coincide with three other points
of pairwise intersection of the circles.


9f

112. Let P denote an arbitrary point of
the circle circumscribed about a rectangle.
Two straight lines passing through the
point P parallel to the sides of the rectangle intersect the sides of the rectangle or
their extensions at points K, L, M, and N.
Prove that N is the intersection point of
the altitudes of the triangle KLM. Prove
also that the feet of the altitudes of the
triangle KLM, different from P, lie on the
diagonals of the rectangle.
113. Drawn in 8 triangle ABC are the
angle hisectorsAD, BE, and CF. The straight
line perpendicular to AD and passing
through the midpoint of AD intersects AC
at a point P. The straight line perpendicular to BE and passing through the midpoint
of BE intersects AB at a point Q. Finally,
the str_ght line perpendicular to CF and
passing' through the midpoint of CF intersects CB at a point R. Prove that the triangles DEF and PQR are equivalent.
t t4. In an isosceles triangle ABC
( I AB I = I BC I ), D is the midpoint of
AC, E the projection of D on BC, F the midpoint of DE. Prove that the lines BF and
AE are mutually perpendicular.
t15. A circle inscribed in a triangle ABC
touches the sides AB and AC at points C1
and B 1 , and the circle touching the side Be
and the extensions of AB and AC touches
the lines AB and AC at points C2 and
B 2 • Let D be the midpoint of the side BC.


92

The line AD intersects the lines BIC I and
B,C t at points E and F. Prove that BECF
is a· parallelogram.
116. A bisector AD of an interior angle is
drawn in a triangle ABC. Let us construct
a tangent l to the circumscribed circle at
a point A. Prove that the straight line
drawn through D parallel to l touches the
inscribed circle of the triangle ABC.
tt7. A straight line is drawn in a triangle
ABC to intersect the sides AC and BC at
points M and N such that I M N I =
I AM I
I BN I Prove that all such
lines touch the same circle.
t18. Prove that the points symmetric to
the centre of the circle circumscribed about
a triangle with respect to the midpoints of
its medians lie on the altitudes of the triangle.
119. Prove that if the altitude of a triangle
is V2 times the radius of. the circumscribed
circle, then the straight line joining the
foot of this altitude on the sides enclosing
it passes through the centre of the circumscribed circle.
120. Let ABC be a right triangle (L C =
90°), CD its altitude, K a point in the
plane such that I AK I = I AC I Prove
that the diameter of the circle circumscribed
about the triangle ABK passing through
the vertex A is perpendicular to the line
DK.

+


93

121. In a triangle ABC a line is drawn
through the vertex A parallel to Be; 8
point D is taken on this line such that
I AD I = I AC I + I AB I; the line segment DB intersects the side AC at a point
E. Prove that- the line drawn through the
point E parallel to BC passes through the
centre of the circle inscribed in the triangle ABC.
122. Two circles pass through a vertex of
an angle and a point lying on the angle
bisector. Prove that the segments of the
sides of the angle enclosed between the circles are congruent.
123. Given a triangle ABC and a point D.
The line AD', BD, and CD for the second
time intersect the circle circumscribed about
the triangle ABC at points AI' Bit and
Cl , respectively. Consider two circles: the
first passes through A and AI' the second
through Band B l . Prove that the end
points of the common chord of these two
circles and the points C and Cl lie on the
same circle.
124. Three parallel lines ll' l2' and Is
are drawn through the vertices A, B, and C
of a triangle ABC, respectively. Prove
that the lines symmetric to i; ll' and ls
with respect to the bisectors of the angles
A, B, and C, respectively, intersect at a
point situated on the circle circumscribed
about the triangle ABC.
125. Prove that if M is 8 point inside a

94


triangle ABC and the lines AM, BM,
and GM pass, respectively, through the centres of the circles circumscribed about the
triangles BMC, CMA, and AMB, then M
is the centre of the circle inscribed in the
triangle ABC.
126. In a triangle ABC points AI' B I ,
and CI are taken on the sides BC, CA, and
AB t respectively. Let M be an arbitrary
point in the plane. The straight line BM
intersects for the second time a circle passing through AI' B, and Cl at a point B t ,
CM intersects the circle described through
AI' B l , and C at a point CI , and AMthe circle passing through A, B l , and C1
at a point At. Prove that the points A 2 ,
Bit Ct t and M lie on the same circle.
127. Let Al be a point symmetric to the
point of tangency of the circle inscribed in
a triangle ABC to the side BC with respect
to the bisector of the angle A. Points R 1
and C1 can be determined in a similar way.
Prove that the lines AAit BRI , eCl' and
the line passing through the centres of the
circles inscribed in and circumscribed about
the triangle ABC meet in the same point.
128. Let AA 1 , BB l t cel be the altitudes
of a triangle ABC. A straight line perpendicular to AB intersects AC and AlGI at
points K and L. Prove that the centre of
KLB 1 lies on the straight line BC.
129. Four circles of equal radius pass


95

through a point A. Prove that three line
segments whose end points are different
from A and are the points of intersection of
two circles (the opposite end points of each
line segment do not belong to one circle)
meet in the same point.
t30. Given a right triangle ABC with a
right angle C. Let 0 be the centre of the circumscribed circle, M the point of tangency
of the inscribed circle and the hypotenuse.
Let a circle centred at M passing through 0
intersect the bisectors of the angles A and
B at points K and L different from O. Prove
that K and L are the centres of the circles inscribed in the triangles ACD and BCD,
respectively, where CD is the altitude of the
triangle ABC.
t3t. Prove that in a triangle A Be the
bisector of the angle A, the midline parallel
to A C, and the straight line joining the
points of tangency of the inscribed circle
with the sides CB and CA intersect at the
same point.
132. Given three straight lines. One of
them passes through the feet of two altitudes
of a triangle, the second line through the
end points of two of its angle bisectors, and
the third through two points at which the
inscribed circle touches the triangle sides
(all the points are situated on two sides of
the triangle). Prove that the three straight
lines intersect at one point.
133. In a triangle ABC points AI' Bit

96


and Cl are taken on the sides Be, CA, and
AB, respectively, such that the lines AA l ,
BB l , and CCI meet in the same point. Prove
that if AA I is the bisector of the angle
BIAICI , then AA I is the altitude of the triangle ABC.
134. Taken on the sides BC, CA, and AB
of a triangle ABC are points AI' B l ,
and CI/f respectively, such that L AAIC =
LBBIA = LeCIB (the angles are measured
in the same direction). Prove that the
centre of the cirle circumscribed about the
triangle bounded by the lines AA l , RB t ,
and eCI coincides with the intersection
135. The vertices of "a triangle AtBtC l
lie on the straight lines Be, CA, and AB
(AI on Be, B l on CA, CIon AB). Prove that
if the triangles ABC and, AIBICI are similar (the vertices A and AI' Band B 1 , C and
CI are similar pairwise), then the intersection point of the altitudes of the triangle
AIBICI is the centre of the circle circumscribed about the triangle ABC. Is the
converse true?
136. Two points are taken on each side of
a triangle such that all the six line segments
joining each point to the opposite vertex are
congruent. Prove that the midpoints of the
six segments lie on the same circle.
t37. In a triangle ABC, line segments
I AM I = I CN I = p are laid off on the
rays AB and CB, where p is the half-pe-

97


rimeter of the triangle (B lies between
A and M, and between C and N). Let K
be a point on the circle circumscribed about
the triangle ABC and diametrically opposite to the point B. Prove that the perpendicular dropped from K on MN passes through
the centre of the inscribed circle.
138. From a point on the circle circumscribed about an equilateral triangle ABC
straight lines are drawn parallel to BC,
CA, and AB which intersect CA, AB, and
Be at points M, N, and Q, respectively.
Prove that M, N, and Q lie on a straight line.
t39. Prove that three lines which are symmetric to an arbitrary straight line passing
through the intersection point of the altitudes of a triangle with respect to the sides of
the triangle are concurrent.
140. Let M be an arbitrary point in the
plane, G the centre of mass of a triangle
ABC. Then the following equality is fulfilled: 3 I MG I 2 = I MA I 2
I MB I 2

+

+

2
I MC I 2 - ~ (I AB 1 + I BC 1 + I CA 1 )

2

2

(Leibniz's theorem).
141. Let ABC be a regular triangle with
side a, and M some point in the plane
found at a distance d from the centre of the
triangle ABC. Prove that the area of the
triangle whose sides are equal to the line
segments MA, MB, and MC can be ex-

pressed by the formula S
7-01 f)57

= ~~3 I a2 _ 3d21.

98


142. Given two regular triangles: ABC
and AIBICt . Find the locus of points M
such that the two triangles formed by the
line segments MA, MB, Me and MAt,
MB 1 , MC I are equivalent.
143. Given a triangle ABC. Line segments
AK and CM are laid off on the rays AB and
CB, respectively, which are equal to AC.
Prove that the radius of the circle circumscribed about the triangle BKM is equal
to the distance between the centres of the
circles circumscribed about and inscribed
in the triangle ABC, and that the straight
line KM is perpendicular to the line joining the centres of the inscribed and circumscribed circles.
144. A straight line is drawn through a
vertex of a triangle perpendicular to the
line joining the centres of the inscribed and
circumscribed circles. Prove that this line
and the sides of the given triangle form two
triangles for which the difference between
the radii of the circumscribed circles is
equal to the distance between the" centres
of the circles inscribed in and circumscribed
about the original triangle.
145. Prove that if the lengths of the sides
of a triangle form an arithmetic progression,
then: (a) the radius of the inscribed circle is
equal to 1/3 of the altitude dropped on the
middle-length side; (b) the line joining the
centre of mass of the triangle and the centre
of the inscribed circle is parallel to the mid-


99

dIe-length side; (c) the bisector of the interior angle opposite to the middle-length
side is perpendicular to the line joining the
centres of the inscribed and circumscribed
circles; (d) for all the points of this angle
bisector, the sum of distances to the sides
of the triangle is constant; (e) the centre of
the inscribed circle, the midpoints of the
largest and smallest sides, and the vertex
of the angle formed by them lie on the same
circle.
146. Let K denote the midpoint of the
side BC of a triangle ABC, M the foot of
the altitude dropped on BC. The circle inscribed in the triangle ABC touches the side
BC at a point D; the escribed circle touching the extensions of AB and AC and the
side BC touches BC at a point E. A common
tangent to these circles, which is different
from the sides of the triangle, intersects the
circle passing through K and M at points
F and G. Prove that the points D, E, F,
and G lie on the same circle.

* * *
147. Prove that the centre of mass of a
triangle, the intersection point of the altitudes, and the centre of the circumscribed
circle lie on a straight line (Euler's* line).
• Euler, Leonhard (1707-1783). A Swiss mathematician.
7*

tOO


148. What sides are intersected by Euler's
line in an acute and an obtuse triangles?
149. Let K denote a point symmetric to
the centre of the cirle circumscribed about a
triangle ABC with respect to the side BC.
Prove that the Euler line of the triangle
ABC bisects the line segment AK.
150. Prove that there is a point P on the
Euler line of a triangle ABC such that the
distances from the centres of mass of the
triangles ABP, BCP, and CAP to the vertices C, A, and B, respectively, are equal.
151. Let P be a point inside a triangle
ABC such that each of the angles APB,
BPC, and CPA is equal to 1200 (any interior angle of the triangle ABC is assumed to
be less than 120°). Prove that the Euler
lines of the triangles APR, BPC, and CPA
meet in the same point.
.,
Remark. When solving this problem use
the result of Problem 296 of this section.
152. Prove that the straight line joining
the centres of the circles inscribed in and
circumscribed about a given triangle is the
Euler line of the triangle with vertices at
the points of tangency of the inscribed circle
with the sides of the given triangle.

• * *
153. Prove that the feet of the perpendiculars from an arbitrary point of the circle
circumscribed about the triangle upon the


101

sides of the triangle are collinear (Simson's·
line).

1M. Prove that the angle between two Simson's lines corresponding to two points of
a circle is measured by half the arc between
these points.
155. Let M be a point on the circle circumscribed about a triangle ABC. A straight
line passing through M and perpendicular
to BC intersects the circle for the second time
at a point N. Prove that the Simson line
corresponding to the point M is parallel to
the line AN.
156. Prove that the projection of the
side AB of a triangle ABC on the Simson
line, corresponding to the point M, is equal
to the distance between the projections of
the point M on the sides A C and BC.
157. Let AA l t BB I , GCI be the altitudes
of a triangle ABC. The lines AA l , BB l t
eCI intersect the circle circumscribed about
the triangle ABC for the second time at
points A 2 , B 2 , C2 , respectively. The Simson
lines corresponding to the points A 2' B 2'
C2 form a triangle A aBaCa (A a is the intersection point of the Simson lines corresponding to the points B 2 and G2 , and so forth).
Prove that the centres of mass of the triangles AIBICI and AaBaGa coincide, while
the lines A 2A a, B 2B a, and C 2C S meet in
the same point.
• Robert Simson (t687-1768). A Scottish mathematician.


102

158. Let AI' B 1 , and CI be points on the
circle circumscribed about a triangle ABC
such that uAA I + vBB I + uCCI =
2kn (all the arcs are measured in the
same direction, k an integer). Prove that the
Simson lines for the triangle ABC corresponding to the points At, B t , and C1 meet
in the same point.
159. Prove"that the tangent to a parabola
at its vertex is a Simson line for a triangle
formed by any three intersecting tangents to
the same parabola.

* • *
160. Prove that the midpoints of the
sides of a triangle, the feet of its altitudes,
and the midpoints, of the line segments between the vertices and the intersection point
of the alti tudes all lie on a circle called the
nine-point circle.
161. Let H denote the ,intersection point
of the altitudes of a triangle, D the midpoint of a side, and K one of the intersection
points of the line HD and the circumscribed circle, D lying between Hand K.
Prove that D is the midpoint of the line
segment HK.
162. Let M denote the median 'point of a
triangle, E the foot of an altitude, F one
of the points of intersection of the line
ME and the circumscribed circle, M lying
between E and F. Prove that I FM I =
21 E M II,


103

163. The altitude drawn to the side BC
of a triangle ABC intersects the circumscribed circle at a point AI. Prove that the
distance from the centre of the nine-point
circle to the side Be is equal to I AA 1 I
164.. In a triangle 4BC, AA 1 is an altitude, H is the intersection point of the altitudes. Let P denote an arbitrary point of
ABC, M a point on the line HP such that
I HP I
I HM I = I HAl I
I HA I
(H lies on the line segment MP if the triangle ABC is acute-angled and outside if
it is obtuse-angled). Prove that M lies on
the nine-point circle of the triangle ABC.
165. In a triangle ABC, BK is the altitude drawn from the vertex B to the side
AC, BL the median drawn from the same
vertex, M and N the projections of the
points A and C on the bisector of the angle
B. Prove that all the points K, L, M,
and N lie on a circle whose centre is located
on the nine-point circle: of the triangle
ABC.
166, Let H be the intersection point of
the altitudes of a triangle, and F an arbitrary point of the circumscribed circle. Prove
that the Simson line corresponding to the
point F passes through one of the intersection points of the line FH and the nine-point
circle (see Problems 153 and 159 of the
section).

t

t04


167. Let l denote an arbitrary line passing
through the centre of the circle circumscribed ahout the triangle ABC, and let At,
B 1 , and C, be the projections of A, B, and
C on l. Three straight lines are drawn:
through Al a line perpendicular to BC,
through B I a line perpendicular to A C,
and through CI a line perpendicular to AB.
Prove that these three lines meet in a point
situated on the nine-point circle of the
triangle ABC.
168. Given a triangle ABC. AA 1 , BB t ,
and CCI are its altitudes. Prove that Euler's lines of the triangles ABICI , AIBCl t
and AlBIC intersect at a point P of the
nine-point circles such that one of the line
segments PA tt PSt, PC t is equal to the sum
of the two others (Thebault's* problem).
169. There are three circles, each of which
passes through a vertex of a triangle and
through the foot of the altitude drawn from
this vertex and touches the radius of the
circle circumscribed about the triangle
which is drawn to this vertex. Prove that
all the circles intersect at two points situated on Euler's line of the given triangle.
170. Consider three circles each of which
passes through one of the vertices of a triangle and through the feet of two angle
bisectors (interior and exterior) emanating
• Thebault, Victor. A modern French geometer.


105

from this vertex (these circles are called
Apollonius's* circles). Prove that: (a) these
three circles intersect at two points (M I
and M I ) ; (b) the line M IM 2 passes through
the centre of the circle circumscribed about
the triangle; (c) the feet of the perpendiculars from the points M I and M 2 upon the
sides of the triangle serve as vertices of two
regular triangles.
171. A straight line symmetric to a median of a triangle about the bisector of the
angle opposite the median is called a symedian. Let the symedian emanating from the
vertex B of a triangle ABC intersect A C
at point K. Prove that I AK I I KC I =

I AB I 2 : I Be I 2.

172. Let D be an arbitrary point on the
side BC of a triangle ABC. Let E and F
be points on the sides AC and AB such that
DE is parallel to AB, and DF is parallel to
A C. A circle passing through D, E, and F
intersects for the second time BC, CA, and
AB at points D I , E I , and F I , respectively.
Let M and N denote the intersection points
of DE and FID I , DF and DIE!, respectively. Prove that M and N lie on the symedian
emanating from the vertex A. If D coincides with the'Ioot of the symedian, then the
circle passing through D, E, and F touches
• Apollonius of Perga (circa 255-170 B.C.).

A great Greek geometer who carried on the work

of Euclid.

106


the side BC. (This circle is called Tucker's·
circle.)
173. Prove that the common chords of
the circle circumscribed about a given triangle and Apollonius' circles are the three
symedians of this triangle (see Problems 170
and 171 of the section).

• * •
174. Given a trapezoid ABCD whose lateral side CD is perpendicular to the 'bases
AD and BC. A circle of diameter AB intersects A'D at a point P (P is different
from A). The tangent to the circle at the
point P intersects CD at a point M. Another
tangent is drawn from M to the circle
touching it at a point Q. Prove that the
straight line BQ bisects CD.
175. Let M and N denote the projections
of the intersection point of the altatudes of a
triangle ABC on the bisectors of the interior and exterior angles B. Prove that the
line MN bisects the side AC.
176. Given a circle and two points A and
B on it. The tangents to the circle which
pass through A and B intersect each other at a
point C. A circle passing through C touches
the line AB at a point B and for the sec• Tucker, Howard Gregory (b. 1922). A modern
American mathematician.


t07

ond time intersects the given circle at a
point M. Prove that the line AM bisects the
line segment CB.
t 77. Drawn to a circle from a point A,
situated outside this circle, are two tangents
AM and AN (M and N the points of tangency) and a secant intersecting the circle at
points K and L. An arbitrary straight line
l is drawn parallel to AM. Let KM and LM
intersect l at points P and Q, respectively.
Prove that the line M N bisects the line segment PQ.
178. A circle is inscribed in a triangle
ABC. Its diameter passes through the point
of tangency with the side BC and intersects
the chord joining two other points of tangency at a point N. Prove that AN bisects
BC.
ABC. Let M be the point at which the circle touches the side AC and M K be the diameter. The line BK intersects AC at a
point N. Prove that I AM I = I NC I
ABC and touches the side BC at a point M,
M K being its diameter. The line A K intersects the circle at a point P. Prove that the
tangent to the circle at the point P bisects
the side BC.
181. A straight line l touches a circle at
a point A. Let CD be a chord parallel to l
and B an arbitrary point on the line l,
The lines CB and DB for the second time

108


intersect the circle at points Land K, respectively. Prove that the line LK bisects the
line segment AB.
182. Given two intersecting circles. Let
A be one of the points of their intersection.
Drawn from an arbitrary point lying on the
extension of the common chord of the given
circles to one of them two tangents touching
it at points M and N. Let P and Q denote
the points of intersection (distinct from A)
of the straigh t lines MA and N A and the second circle, respectively. Prove that the
line MN bisects the line segment PQ.
183. In a triangle ABC, constructed on the
altitude BD as diameter is a circle intersecting the sides AB and BC at points K
and L, respectively. The lines touching the
circle at points K and L intersect at a point
M. Prove that the line BM bisects the side
AC.
.
184. A ~~~aight line l is perpendicular to
the line segment AB and passes through B.
A circle centred on l passes through A and
intersects l at points C and D. The tangents
to the circle at the points A and C intersect
at N. Prove that the line DN bisects the
line segment AB.
triangle ABC. Let N denote the intersection
point of the tangents to the circle which
pass through the points Band C. M is a
point of the circle such that AM is parallel to BC and K is the intersection point


109

of MN and the circle. Prove that KA bisects BC.
186. Let A denote the projection of the
centre of a circle on a straight line l. Two
points Band C are taken on this line such
that I AB I = I AC I . Two arbitrary secants each intersecting the circle at pairs of
points, P, Q and M, N, respectively are
drawn through Band C. Let the lines N P
and MQ intersect the line l at points R
and S, respectively. Prove that I RA I =

IASI

187. Given a triangle ABC. At, B l , Ct
are the midpoints of the sides BC, CA and
AB; K and L are the feet of the perpendiculars from the vertices Band C on the
straight lines AlGI and AIBI , respectively;
o is the centre of the nine-point circle.
Prove that "the line AIO bisects the line
segment KL.

•••
188. Let the points At, B l , Cl be symmetric to a point P with respect to the sides
BC, CA, and AB of a triangle ABC. Prove
that (a) the circles circumscribed about
the triangles AIBC, ABIC, and ABCI have
a common point; (b) the. circles circumscribed about the triangles AlBIC, AIBC I,
and ABtel have a common point.
189. Let AB be the diameter of a semicircle and M a point on the diameter AB.

1fO


Points C, D, E, and F lie on the semicircle
so that LAMD = LEMB, LCMA =
LFMB. Let P denote the intersection
point of the lines CD and EF. Prove that
the line PM is perpendicular to AB.
190. In a triangle ABC, the perpendicular to the side AB at its midpoint D intersects the circle circumscribed about the
triangle ABC at a point E (C and E lie on
the same side of AB), F is the projection
of E on AC. Prove that the line DF bisects
the perimeter.of the triangle ABC, and that
three such lines constructed for each side
of the triangle are concurrent.
191. Prove that a straight line dividing
the perimeter and area of a triangle in
the same ratio passes through the centre
of the inscribed circle.
192. Prove that three lines passing
through the vertices of a triangle and bisecting its perimeter intersect at one point
(called Nagell's* points, Let M denote the
centre of mass of the triangle, I the centre
of the inscribed circle, S the centre of the
circle inscribed in the triangle with vertices
at the midpoints of the sides of the given
triangle. Prove that the points N, M, I,
and S lie on a straight line and I M N I =
2 11M I, I IS I = I SN I·
• Nagell, Trygve (1895-1958). A Norwegian
mathematician.


iff

***
193. Let a, b, and c denote the sides of
a triangle and a + b + c = 2p. Let G
be the median point of the triangle and
0, I and I a the centres of the circumscribed,
inscribed, and escribed circles, respectively (the escribed circle touches the side
BC and the extensions of the sides AB
and AG), R, r, and r a being their radii,
respectively. Prove that the following relationships are valid:
(a) a2 + b2 -f- c2 = 2p 2 - 2r2 - 8Rr;
(b) lOG 12 =

(c)

R2-

~ (a2+b2+c2) ;

1
I IG 12 = 9 (p2+ 5r 2 - 16R r);

(d) 101
(e) lOla

(f) IlIa

12

== R2- 2Rr (Euler);
= R2+ 2Rr a;
= 4R (r a - r).

2
1
2

1

194. Let BB I and GGI denote the bisectors of the angles Band C, respectively ,
of a triangle ABC. Using the notation of
the preceding problem, prove that IBICI 1=
abc
(b+a)(c+a) R

0

I I a I·

195. Prove that the points which are
symmetric to the centr.es of the escribed
circles with respect to the centre of the
circumscribed circle lie on a circle which
is concentric with the inscribed circle

112


whose radius is equal to the diameter of the
circumscribed circle.
tOO. Given a triangle ABC. Prove that
the sum of the areas of the three triangles
the vertices of each of which are the three
points of tangency of the escribed circle
with the corresponding side of the triangle
ABC and the extensions of two other sides
is equal to twice the area of the triangle
ABC plus the area of the triangle with
vertices at the points of tangency of the
circle inscribed in MBC.
t97. Find the sum of the squares of the
distances from the points at which the circle inscribed=in the given triangle touches
its sides to-the centre of the circumscribed
clfrle if the radius of the inscribed circle
i~ r, and that of the circumscribed circle
i~;;R .

t98. A circle is described through the
feet of the angle bisectors in a triangle ABC.
Prove that one of the chords formed by intersection of the circle with the sides of
the triangle is equal to the sum of the other
two chords.
t99. Let AA I , BB l , and CCI be the angle
bisectors of a triangle ABC, L the point of
intersection of the lines AA 1 and BICI,
and K the point of intersection of' the
lines cel and AIBI • Prove tha.t BB I is
the bisector of the angle LBK.
200. In a triangle ABC, points K and
L are taken on the sides AB and Be such


ii3

that I AK I = I KL 1= ILC I. Through
the point of intersection of the lines AL
and CK a straight line is drawn parallel
to the bisector of the angle B to intersect
the line AB at a point M. Prove that
lAM

I = IBG I·

the angle B intersects at a point M the line
passing through the midpoint of AC and
the midpoint of the altitude drawn to AC;
N is the midpoint of the bisector of the
angle B. Prove that the bisector of the
angle C is also the bisector of the angle
MeN.
202. (a) Prove that if a triangle has two
equal angle bisectors then such a triangle
is isosceles (Steiner's theorem).
(b) Prove that if in a triangle ABC t the
bisectors of the angles adjacent to the angles A and C are equal and are either both
inside or both outside the angle ABC, then
I AB I = I BC I. Is it true that, if a triangle has two equal exterior angle bisectors,
then the triangle is isosceles?
203. Given a triangle. The triangle formed by the feet of its angle bisectors is known
to be isosceles. Will the statement that the
given triangle is also isosceles be true?

• Steiner, Jakob (1796-1863). A Swiss mathematician.
8-01557


114

*

* •

204. Let ABCDEF be an inscribed hexagon. Let K denote the' point of intersection
of AC and BF, and L the point of intersection of CE and FD. Prove that the diagonals
AD and BE and the line KL intersect at
the same point (Pascal's theorem).
205. Given a triangle ABC and a point
M. A straight line passing through the
point M intersects the lines AB, BC, and
CA at points Ct , At, and B t , respectively.
The lines AM, BM, and CM intersect the
circle circumscribed about the triangle ABC
at points A 2 , B 2 , and C2 , respectively. Prove
that the lines A tA 2 , B tB 2 , and CtC 2 intersect at a point situated on the circle circumscribed about the triangle ABC.
206. Two mutually perpendicular lines
are drawn through the intersection point
of the altitudes of a triangle. Prove that
the midpoints of the line segments intercepted by these lines on the sides of the
triangle (that is, on the lines forming the
triangle) lie on a straight line.

**

*

207. Given a triangle ABC and an arbitrary point P. The feet of the perpendiculars dropped from the point P on the
sides of the triangle ABC serve as the vertices of the triangle AIBIC I. The vertices


t15

of the triangle A 2 8 2 C2 are found at the
intersection points (distinct from A, B, C)
of the straight lines AP, BP, and CP
with the circle circumscribed about the
triangle ABC. Prove that the triangles
AIBlCl and A 2B 2C t are similar. How many
points P are there for a scalene triangle
ABC such that the corresponding triangles
AIBIC l and A 2B 2C 2 are similar to the triangle ABC?
208. Let At,
Cl denote the feet of
the. perpendiculars dropped from an arbitrary point M on the sides BC, CA, and
AB of a triangle ABC, respectively. Prove
that three straight lines passing through
the midpoints of the line segments BlC l
and MA, CIA 1 and ME, AlB l and MC
intersect at one point.
209. Let 8 be the area of a given triangle,
and R the radius of the circle circumscribed
about this triangle. Let, further, 8 1 denote
the area of the triangle formed by the feet
of the perpendiculars dropped on the sides
of the given triangle from a point located
at a distance d from the centre of the cir81 =
cumscribed circle. Prove that

e;

'~

I ~ - ~t

(Euler's theorem).

210. Prove that if A, B, C, and Dare
arbitrary points in the plane, then the
four circles each of which passes through
the midpoints either of' the line 'Segment~
AR, AC, and AD·; or BA, BC, and BD;
8*

1t6

Problems 'in Plane Geometry

or CA, CB, and CD; or DA, DB, and DC
have a common point.
21t. Given a triangle A BC and an arbitrary point D in the plane. The triangle
formed by the feet of the perpendiculars
dropped from D on the sides of the triangle
ABC will be called the pedal triangle of the
point D with respect to the triangle ABC, and
the circle circumscribed about the pedal
triangle, the pedal circle. Let D I denote
the point of intersection of the lines symmetric to the lines AD, BD, and CD with
respect to the bisectors of the angles A, B,
and C, respectively, of the triangle ABC.
Prove that the pedal circles of the points D
and D 1 coincide.
212. Consider four points in the plane no
three of which are collinear. Prove that
the' four pedal circles each of which corresponds to one of the points under consideration with respect to the triangle whose
vertices are the remaining three points
213. A straight line passing through the
centre of the circle circumscribed about a
triangle ABC intersects AB and AC at
points C1 and B 1 , respectively. Prove
that the circles constructed on BB I and CCI
as diameters intersect at two points one of
which lies on the circle circumscribed about
the triangle ABC, the other on the ninepoint circle of the. triangle ABC.


117

Quadrilaterals
214. Given an inscribed quadrilateral
ABCD. The circle diameter is AB. Prove
that the projections of the sides AD and BC
on the line CD are equal in length.
215. In a convex quadrilateral ABeD:
o is the point of intersection of its diagonals, E, F, and G are the projections of B,
C, and 0 on AD. Prove that the area of
the quadrilateral is equal to I AD f~~·,' CF I.
216. Let ABCD be a convex quadrilateral. Consider four circles each of which
touches three sides of this quadrilateral.
(a) Prove that the centres of these circles
lie on one circle.
(b) Let r 1 , r 2 , r a, and r 4 denote the radii
of these circles (r 1 does not touch the side
DC, r 2 the side DA, r, the side AR, and r,
.IABI
ICDI
the side BC). Prove that - - + -- =
rl
r3
IBGI + IADI.

'.J

r2

rt

217. Prove that for the area S of an inscribed quadrilateral the following formula holds true:
S = V(p - a) (p - b) (p - c) (p - d),
where p is the semiperimeter, and a, b, C,
and d are the sides of the quadrilateral.
218. Let 2cp be the sum of two opposite
angles of a circumscribed quadrilateral,
a, b, c, and d its sides, S its area. Prove
that S = Yabcd sin cp.

118


219. Points M and N are taken on the
sides AB and CD of a convex quadrilateral
ABCD to divide them in the same ratio
(counting from the vertices A and C). J oining these points to all the vertices of the
quadrilateral ABCD, we separate the latter
into six triangles and a quadrilateral. Prove
that the area of the quadrilateral thus obtained is equal to the sum of the areas of
two triangles adjacent to the sides fiC and
AD.
220. A diameter AB and a chord CD
which does not intersect that diameter are
drawn in a circle. Let E and F denote
the feet of the perpendiculars dropped
from the points A and B on the line CD.
Prove that the area of the quadrilateral
AEFB is equal to the sum of the areas of
the triangles ACB and ADB.
221. Given a convex quadrilateral QlFour straight lines perpendicular to its
sides and passing through their midpoints
form a quadrilateral Q2. A quadrilateral
Q3 is formed in the same way for the quadrilateral 02- Prove that the quadrilateral
Q3 is similar to the original quadrilateral

01-

222. Points M and N are taken on opposite sides Be and DA of a convex quadrilateral such that I BM I : I MC I = IAN I
I ND I = I AB I : I CD I. Prove that the
line M N is parallel to the bisector of the
angle formed by the sides AB and CD_


1i9

223. A convex quadrilateral is separated
by its diagonals into four triangles. The
circles inscribed in these triangles are of
tile same radius. Prove that the given quadrilateral is a rhombus.
224. The diagonals of a quadrilateral separate the latter into four triangles having
equal perimeters. Prove that the quadrilateral is a rhombus.
225. In a quadrilateral ABeD, the circles inscribed in the triangles ABC, BCD,
CDA, DAB are of the same radius. Prove
that the given quadrilateral is a rectangle.
226. A quadrilateral ABeD is inscribed
in a circle. Let M be the point of intersection of the tangents to the circle passing
through A and C, N the point of intersection of the tangents drawn through Band
D, K the intersection point of the bisectors
of the angles A and C of the quadrilateral,
L the intersection point of the angles Band
D. Prove that if one of the four statements
is true, i.e.: (a) M belongs to the straight
line BD, (b) N belongs to the straight line
AC, (c) K lies on BD, (d) L lies on AC,
then the remaining three statements are
also true.
227. Prove that four lines each of which
passes through the feet of two perpendiculars dropped from a vertex of an inscribed
quadrilateral on the sides not including this
vertex intersect at one point.
228. Let AB and CD be two chords of

120


a circle, M the point of intersection of
two perpendiculars: one of them to AB at
the point A and the other to CD at the
point C. Let N be the point of intersection of the perpendiculars to AB and CD
at the points Band D, respectively. Prove
that the line MN passes through the point
of intersection of BC and AD.
circle of radius R passes through the points
A and B. Another circle of the same radius
passes through the points Band C. Let
M denote the second point of intersection
of these circles. Prove that the radii of the
AMD and CMD are R.
230. Let ABeD be a parallelogram. A
circle touches the straight lines AB and AD
and intersects BD at points M and N.
Prove that there is a circle passing through
M and N and touching the lines eB and
CD.
231. Let ABCD be a parallelogram. Let
us construct a circle on the diagonal AC as
diameter and denote by M and N the points
of intersection of the circle with the lines
AB and AD, respectively. Prove that the
lines BD and MN and the tangent to the
circle at the point C intersect at the same
point.
232. A quadrilateral ABeD is inscribed
in a circle. Let 01' 02' Os,
be the centres of the circles inscribed in the triangles

o,


121

ABC, BCD, CDA, DAB, respectively, and
HI' H t , H 3 , and H 4 the intersection points
of the altitudes of the same triangles. Prove
that the quadrilateral 0 101030 4 is a rectangle, and the quadrilateral HIH 2 HsH c is equal
to the quadrilateral ABCD.
233. Given a triangle ABC and an arbitrary point D in the plane. Prove that
the intersection points of the altitudes
of the triangles ABD, BCD, CAD are the
vertices of the triangle equivalent to the
given one.
234. Prove that if a circle can be inscribed in a quadrilateral, then: (a) the
circles inscribed in the two triangles into
which the given quadrilateral is separated
by a diagonal touch each other, (b) the
points of tangency of these circles with the
sides of the quadrilateral are the vertices
of the inscribed quadrilateral.
235. Prove tqat if ABCD is an inscribed
quadrilateral, then the sum of the radii
of the circles inscribed in the triangles ABC
and ACD is equal to the sum of the radii
inscribed in triangles BCD and RDA.

•**
236. Let a, b, c, and d be the sides of a
quadrilateral, m and n its diagonals, A
and C two opposite angles, Then the following relationship is fulfilled: m 2n2 = a 2c 2
bltP - 2abcd cos (A + C) (Bretschneider's

+

122


theorem or the law of cosines for a quadrilateral).
237. Let a, b, C, and d denote the sides Qf
an inscribed quadrilateral and m and n
its diagonals. Prove that mn = ac
bd
(Ptolemy's· theorem).
238. Prove that if ABC is a regular
triangle, M an arbitrary point in the plane
not lying on the circle circumscribed about
the triangle ABC, then there is a triangle
whose sides are equal to I MA ), I MB I,
and I MC I (Pompeiu's** theorem). Find
the angle of this triangle which is opposite
the side equal to I MB I if LAMC = ct.
239. Let ABCD be an inscribed quadrilateral. Four circles, (x, ~, "(, and i), touch
the circle circumscribed about the quadrilateral ABCD at points A, B, C, and D,
respectively. Let t a f3 denote the segment
of the tangent to the circles ex and ~, t(J,~
being the segment of a common external
tangent if ex and ~ touch the given circle
in the same manner (internally or externally), and the segment of a common in.
ternal tangent if ct and ~ touch the given
circle in a different way (the quantities
t~'V, t a 6 , etc. are defined in a similar way).

+

• Ptolemy (Caudius Ptolemaus) (circa A.D.
150). An Alexandrian geometer, astronomer. and
geogra pher .
• ~. Pompeiu, Dirnitrie (1873-1954). A Rumanian
mathematician.


123

Prove that
tcsptyr,

+ tf,ytr,~

= tayt{ir,

(*)

(Ptolemy's generalized theorem).
240. Let cx, p, "I, and ~ be four circles in
the plane. Prove that if the following rela tionshi p is fulfilled:
t a (3t YlJ + tf.,y ttH1. = tavtf,t"
(*)
where taf.,' etc. are line .segments of common external or internal tangents to the
circles cx and ~, etc. (for any three circles
we take either three external tangents or
one external and two internal, then the circles cx, p, "I, and 6 touch the same circle.

*•*
241. The extensions of the sides AB and
DC of a convex quadrilateral ABCD intersect at a point K, and the extensions
of the sides AD and Be at a point L, the
line segment BL intersecting DK. Prove
that if one of the three relationships

lAB 1 + 1 CD 1 = 1 Be 1 + 1 AD I,
IBK 1+ IBL 1"= IDK I + IDL I,
IAKI+ICLI=IALI+ICKI
is fulfilled, then the two others are also fulfilled.
242. The extensions of the sides AB and
DC of a convex quadrilateral ABCD intersect at a point K, and those of the sides

124


AD and BC at a point L, the line segment
BL intersecting DK. Prove that if one of
the three relationships 1 AD I + I DC I =
I AB I I CB I, 1AK I I CK I =
I AL I~-+ 1CL I, I' BK I I DK I =
I BLI
I DL I is fulfilled, then the two
others are also fulfilled.
243. Prove that if there exists a circle
touching the straight lines AR, BC, CD,
and DA, then its centre and the midpoints
of A C and BD are collinear.
244. Let ABeD be an inscribed quadrilateral. The perpendicular to BA erected
at a point A intersects the line CD at a
point M, the perpendicular to DA erected
at A intersects the line Be at a point N.
Prove that MN passes through the centre
of the circle circumscribed about the quadrilateral ABeD .
.245. Let ABCD be an inscribed quadrilateral, E an arbitrary point on the
straight line AB, and F an arbitrary point
on the line DC. The straight line AF intersects the circle at a point M, and the line
DE at a point N. Prove that the lines
Be, EF, and MN are either concurrent or
parallel.
246. Prove that the feet of the perpendiculars dropped from the intersection point
of the diagonals. of an inscribed quadrilateral on its sides are the vertices of a quadrilateral in which a circle can be inscribed.
Find the radius of that circle if the diago-

+
+

+
+


125

nals of the inscribed quadrilateral are mutually perpendicular, the radius of the given
circle is R, and the distance from its centre
to the point of intersection of the diagonals
is d. .
247. The diagonals of an inscribed quadrilateral are mutually perpendicular. Prove
that the midpoints of its sides and the
point of intersection of the diagonals on
the sides lie on a circle. Find the radius
of that circle if the radius .of the gi ven
circle is R, and the distance from its centre
to the point of intersection of the diagonals
of the quadrilateral is d.
248. Prove that if a quadrilateral is both
inscribed in a circle of radius R and circumscribed about a circle of radius r, the
distance between the centres of those circles being d, then the relationship

~ d)1

(R~d)t+

1Ir2 is true. In this case
there are infinitely
many quadrilaterals both inscribed in the larger circle
and circumscribed about the smaller one
(any point of the larger circle may be taken
as one of the vertices).
249. A convex quadrilateral is separated
by its diagonals into four triangles. Prove
that the line joining the centres of mass
of two opposite triangles is perpendicular
to the straight line connecting the inter(R

t26


section points of the altitudes of two other
triangles.
250. Let ABCD be an inscribed quadrilateral, M and N the midpoints of AC
and BD, respectively. Prove that if BD
is the bisector of the angle ANC, then AC
is the bisector of the angle BMD.
251. Let ABCD be an inscribed quadrilateral. When extended, the opposite
sides AB and CD intersect at a point K,
and the sides BC and AD at a point L.
Prove that the bisectors of the angles BKC
and BLA are mutually perpendicular and
their intersection point lies on the straight
line joining the midpoints of AC and ED.
252. The diagonals of a quadrilateral are
mutually perpendicular. Prove that the
four straight lines each of which joins
one of the vertices of the quadrilateral to
the centre of the circle passing through
that vertex and two adjacent ones of the
quadrilateral intersect at one point.
253. Let P, Q, and M are the respective
intersection points of the diagonals of an
inscribed quadrilateral and the extensions
of its opposite sides. Prove that the intersection point of the altitudes of the triangle
PQM coincides with the centre of the circle
circumscribed about the given quadrilateral
iBrodcard's theorem).
254. Let ABCD be a circumscribed quadrilateral, K the 'point of intersection of the
straight lines AB and CD, L the point of in-


t27

tersection of AD and Be. Prove that the
intersection point of the altitudes of the
triangle formed by the lines KL, AC, and
BD coincides with the centre of the circle
inscribed in the quadrilateral ABeD.
255. Let ABeD be a convex quadrilateral, LABC = LADC, M and N the feet
of the perpendiculars dropped from A on
Be and CD, respectively, K the point of
intersection of the straight lines MD and
NB. Prove that the straight lines AK and
MN are mutually perpendicular.

*• *
256. Prove that four circles circumscribed about four triangles formed by four
intersecting straight lines in the plane
have a common point (Michell's· point).
257. Prove that the centres of four circles
circumscribed about four triangles formed
by four intersecting straight lines in the
plane lie on a circle.
258. Given four pairwise intersecting
lines. Let M denote the Michell's point
corresponding to these lines (see Problem
256 of Sec. 2). Prove that if four of the six
points of pairwise intersection of the given
lines lie on a circle centred at 0, then the
straight line passing through the two re• Michell, John Henry (i879-t940). An Australian mathematician.

128


maining points contains the point M and
is perpendicular to OM.
259. Four pairwise intersecting straight
lines form four triangles. Prove that if
one of the lines is parallel to Euler's line
(see Problem 147 of Sec. 2) of the triangle
formed by the three other lines then any
other line possesses the same property.
260. Given a triangle ABC. A straight
line intersects the straight lines AB, Be,
and CA at points D, E, and F, respectively.
The lines DC, AE, and BF form a triangle
KLM. Prove that the circles constructed
on DC, AE, and BF as diameters intersect
at two points P and N (these circles are
assumed to intersect pairwise), and the line
PN passes through the centre of the circle
circumscribed about the triangle KLM and
also through the intersection points of the
altitudes of the triangles ABC, BDE, DAF,
and CEF.
261. Given a triangle ABC. An arbitrary line intersects the straight lines AB, Be,
and CA at points D, E, and F, respectively.
Prove that the intersection points of the
altitudes of the triangles ABC, BDE, DAF,
and CEF lie on one line perpendicular to
the Gaussian line (see Problem 53 of Sec. 2).
262. Prove that the middle perpendiculars to the line segments joining the intersection points of the altitudes to the centres of the circumscribed circles of the
four triangles formed by four arbitrary


f29

straight lines in the plane intersect at one
point (Herwey's point).
263. Consider sixteen points serving as
centres of all possible inscribed and escribed
circles for four triangles formed by four
intersecting lines in the plane. Prove that
these sixteen points can be grouped into
four quadruples in two ways so that each
quadruple lies on one circle. When the first
method is used the centres of these circles
lie on one line, when the second-on the
other line. These lines are mutually perpendicular and intersect at Michell's point,
which is a common point of the circles circumscribed about four triangles.

Circles and Tangents.
Feuerbach's Theorem
264. On a straight line, points A, B, C,
and D are situated so that I BC I = 2 I AB I,
I CD I = I AC I. One circle passes through
the points A and C, and the other through
the points Band D. Prove that the common
chord of these circles bisects the line segment AC.
265. Let B denote a point belonging to
the line segment AC. The figure bounded
by the arcs of three semicircles of diameters AR, BC, and CA lying on the same
side of the line A C is called the shoemaker
knife or Archimedean arbelos. Prove that
the radii of two circles each of which
9-015$7

130


touches both two semicircles and the line
perpendicular to A C and passing through
B are equal to each other (Archimedean
problem).

266. Of three circles each passes through
two given points in the plane. Let 01' 02'
Os denote their centres. The straight line
passing through one of the points common
to all the three circles intersects them for
the second time at points AI' A 2 , As, respectively. Prove that I A 1A 2 I I A 2A s I =
I 0 10 2 I : I 0 20 3 I·
267. Given two non-intersecting circles.
Prove that the four points of tangency of
common external tangents to these circles
lie on a circle; in similar fashion, the
four points of tangency of common internal
tangents lie on a second circle, and the four
points of intersection of the common internal tangents with the common external tangents lie on a third circle, all the three
circles being concentric.
268. Given two non-intersecting circles.
A third circle touches them externally and
is centred on the line passing through the
centres of the given circles. Prove that the
third circle intersects the common internal
tangents to the given circles at four points
forming a quadrilateral two sides of which
are parallel to the common external tangents
to the given circles.
269. Given two circles. A straight line
intersecting one circle at points A and C


t3t

and the other at points Band D is drawn
through the centre of the first circle. Prove
that if I AB I I BC I = I AD I : I DC I,
then the circles are perpendicular, that
is, the angle between the tangents to them
at the point of their intersection is a right
one.
270. Points A, B, C, and D lie on a
circle or a straight line. Four circles are
drawn through the points A and B, B
and C, C and D, D and A. Let B I , CI , D 1 ,
and Al denote the intersection points (distinct from A, B, C, and D) of the first and
second, the second and third, the third and
fourth, the fourth and first circles, respectively. Prove that the points AI' B 1 , CI ,
and D 1 lie on a circle (or a straight line).
271. From a point A taken outside a
circle, two tangents AM and AN (M and
N points of tangency) and two secants are
drawn. Let P and Q denote the intersection points of the circle with the first secant, and K and L with the second one, respectively. Prove that the straight lines
PK, QL, and MN either intersect at a
point or are parallel.
Try to develop the method of construction of a tangent to a given circle through
a given point with a ruler alone.
272. ·Given a circle with centre 0 and a
point A. Let B denote an arbitrary point
of the circle. Find the locus of intersection
points of tangents to the circle at the point
9*

132

Problema in Plane Geometry

-B with the straight line passing through

o perpendicular to AB.

273. Given a circle and two points A and
B on it. Let N be an arbitrary point on
the line AB. We construct two circles, each
passing through the point N and touching
the given circle: one at a point A, the other
at a point B. Let M denote a second point
of intersection of those circles. Find the
locus of points Al.
.
274. Two arbitrary chords PQ and KL
are drawn through a "fixed point inside
a circle. Find the locus of intersection points
of the lines P K and QL.
275. Two circles intersect at points A
and B. An arbitrary straight line passes
through the point B and, for the second time,
intersects the first circle at a point C,
and the second at a point D. The tangents
to the first circle at C and to the second at
D intersect at a point M. Through the
point of intersection of AM and CD, there
passes a line parallel to CM and intersecting AC at a point K. Prove that KB touches
the second circle.
276. Given a circle and a tangent l to
it. Let·N denote the point of tangency,
and NM the diameter. On the line NM
a fixed point A is taken. Consider an arbitrary circle passing through the point
A with centre on l. Let C and D be the points
of intersection of this circle with l, and P
and Q the points of intersection of the


133

straight lines MC and MD with the given
circle. Prove that the chord PQ passes
through the fixed point in the plane.
277. The points 0 1 and O2 are the centres
of two intersecting circles, A being one of
the points of their intersection. Two common tangents are drawn to the circles; BC
and EF are the chords of those circles
with ends at the points of tangency (C and
F being most remote from A), M and N
are the midpoints of BC and EF, respectively. Prove that L01AOt = LMAN =
2 LCAE.
278. A diameter AB is drawn in a circle,
CD being a chord perpendicular to AB.
An arbitrary circle touches the chord CD
and the arc CBD. Prove that a tangent to this
circle drawn from the point A is equal to A C.
279. Given a segment of a circle. Two
arbitrary circles touch the chord and the
arc .of the segment and intersect at points
M and N. Prove that the straight line MN
passes through a fixed point in the plane.
280. Given two equal non-intersecting
circles. Two arbitrary points F and F' are
taken on two common internal tangents.
From both points one more tangent can
be drawn to each of the circles. Let the tangents drawn from the points F and F ' to
one circle meet in a point A, to the other
in a point B. It is required to prove that:
(1) the line AB is parallel to the line joining the centres of the circles (in the case

i34


of unequal circles, it passes through the
intersection point of the external tangents);
(2) the line joining the midpoints of FF'
and AB passes through the midpoint of
the line segment joining the centres of the
circles.
(This problem was suggested to the readers of "The Bulletin of Experimental Physics and Elementary Mathematics" by Professor V. Ermakov. This journal was issued in Russia last century. The problem
WlS published in issue 14(2) of "The Bulletin" in 1887. A prize, some mathematical
books, was offered to readers for the correct
solution.)
281. Given three circles a, p, and 1.
Let II and l2 denote the common internal
tangents to the circles ex and p, mt and m 2
the common internal tangents to the circles
p and "I, and n1 and n 2 to the circles y and
a. Prove that if the lines ll' ml' and n l are
concurrent, then the lines l2' m 2 , and n 2
are also concurrent.
282. An arc AB of a circle is divided
into three equal parts by the points C and
D (C is nearest to A). When rotated about
the point A through an angle of n/3, the
points B, C, and D go into points B I , CI ,
and D 1 ; F is the point of intersection of
the straight lines AB I and DC I ; E is a
point on the bisector of the angle B1BA
such that I BD I = I DE I. Prove that the
triangle CEF is regular (Finlay'S theorem).


* * *

135

283. Given an angle with vertex A and
a circle inscribed in it. An arbitrary straight
line touching the given circle intersects the
sides of the angle at points Band C. Prove
that the circle circumscribed about the
triangle ABC touches the circle inscribed
in the given angle.
284. In a triangle ABC, a point D is
taken on the side AC. Consider the circle
touching the line segment AD at a point
M, the line segment BD and the circle
circumscribed about the triangle ABC.
Prove that the straight line passing through
the point M parallel to BD touches the
circle inscribed in the triangle ABC.
285. In a triangle ABC, a point D is
taken on the side AG. Let 0 1 be the centre
of the circle touching the line segments AD,
BD, and the circle circumscribed about
the triangle ABC, and let O2 be the centre
of the circle touching the line segments CD,
BD, and the circumscribed circle. Prove
that the line 0 102 passes through the centre
of the circle inscribed in the triangle
ABC and I 0 10 I I 00 2 I = tan'' (q>/2),
where q> = LBDA (Thebault's theorem).
286. Each of four circles touches internally a given circle and two of its mutually
intersecting chords. Prove that the diagonals of the quadrilateral with vertices
at the centres of those Iour circles are mutually perpendicular.

o

t36


•••
287. Prove that the nine-point circle
(see Problem 160 of Sec. 2) touches the
circle inscribed in the triangle and all of
the escribed circles (Feuerbach's theorem).
288. Let H denote the intersection point
of the altitudes of a triangle ABC. Prove
that the nine-point circle touches all of
the inscribed and escribed circles of the
triangles AHB, BHC, and CHA~
289. Prove that the intersection point
of the diagonals of the quadrilateral with
vertices at the points of tangency of the
nine-point circle of a triangle ABO with the
inscribed and escribed circles of the triangle
lies on its midline.
290. Let F, F (u
and Fe denote the
points of tangency of the nine-point circle
of a triangle ABC with the inscribed and
three escribed circles (F a is the point of
tangency with the circle centred at I a and
so on). Let further Al and AI' B 1 "and BIt
and C1 and C2 denote the intersection points
of the bisectors of the interior and exterior
angles A, B, and C with the opposite
sides, respectively. Prove that the following triangles are similar pairwise: 6F a,F b F c
and ~lBIClt 6FF bF e and ~lBIC2'
AFFeFo and .6.B1 C1A 1 t 6FFa,Fb and
l:i.C1A 1B a (Thebault's theorem).

r;

See. 2. Selected Problems

t37

Combinations of Figures. Displacements
in the Plane. Polygons

29t. Three squares BCDE, ACFG, and
BAHK are constructed externally on the
sides BC, CA, and AB of a triangle ABC.
Let FCDQ and EBKP be parallelograms.
Prove that the triangle APQ is a right
isosceles triangle.
292. Let ABeD be a rectangle, E a point
on BC, F a point on DC, E 1 the midpoint
of AE1 , F 1 the midpoint of AF. Prove that
if the triangle AEF is equilateral, then the
triangles DEle and BF1C are also equilateral.
293. Two squares ACKL and BCMN are
constructed externally on the legs AC and
Be of a right triangle. Prove that the quadrilateral bounded by the legs of the given
triangle and the straight lines LB and
NA is equivalent to the triangle formed
by the lines LB, N A, and the hypotenuse
AB.
294. Squares are constructed externally
on the sides of a convex quadrilateral.
Prove that if the diagonals of the quadrilateral are mutually perpendicular, then
the line segments joining the centres of
the opposite squares pass through the intersection point of the diagonals of the
quadrilateral.
295. Prove that if the centres of the
squares constructed externally on the sides

t38


of a given triangle serve as the vertices
of the triangle whose area is twice the area
of the given triangle, then the centres of
the squares constructed internally on the
sides of the triangle lie on a straight line.
296. Constructed externally on the sides
BC, CA, and AB of a triangle ABC are
trianglesA 1BC, BICA, and CtAB such that
LAIBC = LCIBA,
LC1AB = LB1AC,
LB 1CA = LA leB. Prove that the lines
AA I , BB1 , eCI intersect at a point.
297. Let ABC be an isosceles triangle
(I AB 1= I BC I) and BD its altitude.
A disc of radius BD rolls along the straight
line AC. Prove that as long as the vertex
B is inside the disc, the length of the circular arc inside the triangle is constant.
298. Two points move in two intersecting
straight lines with equal velocities. Prove
that there is a fixed point in the plane
which is equidistant from the moving
points at all instants of time.
299. Two cyclists ride round two intersecting circles, each funning round his
circle with a constant speed. Having
started simultaneously from a point at
which the circles intersect, the cyclists
meet once again at this point after one
circuit. Prove that there is a fixed point
such that the distances from it to the cyclist are equal all the time if they ride: (a)
in the same direction (clockwise); (b) in
opposite directions.

139

300. Prove that: (a) the rotation about
a point 0 through an angle ex is equivalent
to two successive axial symmetry mappings whose axes pass through the point
and the angle between the axes is a/2;
8 translation is equivalent to two axial
symmetry mappings with parallel axes;
(b) two successive rotations in the same direction, one about the point 0 1 through an
angle ex, and the other about the point O 2
through an angle p (0 ~ a < 2n, 0 ~ p <
2n) are equivalent to one rotation through
an angle ex + p about a certain point 0
if ex,
p =1= 2n. Find the angles of the
triangle 0 1020.
301. Given an arbitrary triangle ABC.
Three isosceles triangles A KB, BLC, and
CMA with the vertex angles K, L, and M
equal to a, p, and "I, respectively, a
p
"I = 2n are constructed on its sides as bases.
All the triangles are located either outside the triangle ABC or inside it. Prove
that the angles of the triangle KLM are
equal to a'/2, P/2, "1/2.
302. Let ABCDEF be an inscribed
hexagon
in· which I AB I = I CD I =
I EF I = R, where R is the radius of the
circumscribed circle, 0 its centre. Prove
that the points of pairwise intersections of
the circles circumscribed about the triangles
BOC, DOE, FOA, distinct from 0, serve as
the vertices of an equilateral triangle with
side R.

°

+

+ +

t40


303. Four rhombi each having an acute
angle a are constructed externally on the
sides of a convex quadrilateral. The angles
of two rhombi adjacent to one vertex of
the quadrilateral are equal. Prove that
the line segments joining the centres of
opposite rhombi are equal to each other,
and the acute angle between those segments
is a.
304. Given an arbitrary triangle. Constructed externally on its sides are equilateral triangles whose centres serve as vertices of the triangle d. The centres of the
equilateral triangles constructed internally
on the sides of the original triangle serve
as vertices of another triangle 6. Prove that:
(a) d and 6 are equilateral triangles; (b)
the centres of d and 6 coincide with the
centre of mass of the original triangle; (c),
the difference between the areas of ~ and ()
is equal to the area of the original triangle.
305. Three points are given in a plane.
Through these points three lines are drawn
forming a regular triangle. Find the locus
of centres of those triangles.
306. Given a triangle ABC. On the line
passing through the vertex A and perpendicular to the side BC, two points Al and
AI are taken such that I AA 1 I = I AA 2 I =
I Be I (AI being nearer to the line BC
than A 2). Similarly, on the line perpendicular to AC and passing through B points
8 1 and B 2 are taken such that I BB1 I =


t4i

I BB 2 I = I AC I. Prove that the line segments AlB,. and A 2Bl are equal and mutually perpendicular.
* • lie
307. Prove that a circumscribed polygon
having equal sides is regular if the number of its sides is odd.
308. A straight line is drawn through
the centre of a regular n-gon inscribed in
a unit circle. Find the sum of the squares
of the distances from the line to the vertices
of the n-gon.
from an arbitrary point inside a convex
polygon to its sides is constant if: (8) all
the sides of the polygon are equal; (b)
all the angles of the polygon are equal.
310. A semicircle is divided by the
points A o' AI' .. , A S n +1 into 2n + 1
equal arcs (A o and A t n +1 the end' points
of the semicircle), 0 is the centre of the
semicircle. Prove that the straight lines
AlAin, A.Aln-l'. ., A nA n+1 t when intersecting the straight lines OA nand OA n +1 ,
form line segments whose sum is equal
to the radius of the circle.
31 t. Prove that if perpendiculars are
drawn to the sides of an inscribed 2n-gon
form an arbitrary point of a circle, then the
products of the lengths of the alternate
perpendiculars are equal.
312. Let AlAI ... An be an inscribed

142


polygon; the centre of the circle is found
inside the polygon. A system of circles
touch internally the given circle at points
AI' At,. ., An' one of the intersection
points of two neighbouring circles lying
on a side of the polygon. Prove that if n
is odd, then all the circles have the same
radius. The length of the outer boundary
of the union of the inscribed circles is
equal to the circumference of the given
circle.
313. Consider the circle in which a regular (2n + i)-gon A 1A 2
A 2 n + 1 is inscribed. Let A be an arbitrary point of the
arc A 1A 2 n + 1 •
(a) Prove that the sum of the distances
from A to the even vertices is equal to the
sum of the distances from A to the odd vertices.
(b) Let us construct equal circles touching the given circle in the same manner at
points AI' A 2 ,
., A t n + 1 • Prove that
the sum of the tangents drawn from A to
the circles touching the given circle at even
vertices is equal to the sum of the tangents
drawn to the circles touching the given
circle at odd vertices.
314. (a) Two tangents are drawn to a
given circle. Let A and B denote the points
of tangency and C the point of intersection of the tangents. Let us draw an arbitrary straight line l which touches the given
circle and does not pass through the points


t43

A and B. Let u and v be the distances from
A and B to l, respectively, w the distance
from C to I. Find uv/w2 if LACB = cx.
(b) A polygon is circumscribed about
a circle. Let I be an arbitrary line touching

the circle and coinciding with no side of the
polygon. Prove that the ratio of the product of the distances from the vertices of
the polygon to the line l to the product of
the distances from the points of tangency
of the sides of the polygon with the circle
to I is independent of the position of the
line I.
(c) Let A IA 2
A 2 n be a 2n-gon circumscribed about a circle and l an arbitrary tangent to the circle. Prove that the
product of the distances from the odd vertices to the line l and the product of the
distances from the even vertices to the line
l are in a constant ratio independent of
l (the line I is assumed to contain no vertices of the polygon).
315. Drawn in an inscribed polygon are
non-intersecting diagonals separating the
polygon into triangles. Prove that the sum
of the radii of the circles inscribed in those
triangles is independent of the way the
diagonals are drawn.
316. Let A IA 2 • An be a polygon
of perimeter 2p circumscribed about a
circle of radius r, B I , B 2 ,
., B n the
points at which the circle touches the
sides A IA 2 , A 2A s , · · ., AnAl, respective-

144


ly, and M a point found at a distance d
from the centre of the circle. Prove that
2
IMB lI2·IA 1A 1 1+ IMB~ 1 · IA 2 A s 1+.
+
2.1
1MB n 1 AnAl I = 2p (r2 + d 2 ) .
317. Let ABCD denote an inscribed
quadrilateral, M an arbitrary point on the
circle. Prove that the projections of the
point M on Simson's lines (see Problem
153 of Sec. 2), corresponding to the point M
with respect to the triangles ABC, BCD,
CDA, and DAB, lie in a straight line (Simson's line of a quadrilateral).
Further, knowing Simson's line of an
n-gon, let us determine Simson's line of
an (n + i)-gon by induction. Namely,
for an arbitrary inscribed (n + i)-goll
and a point M on the circle, the projections of this point on all possible Simson's
lines of this point with respect to all possible n-gons formed by n vertices of this
(n + 1)-gon lie on a straight line which
is Simson's line of an (n + i)-gone
318. A circle ~ is situated inside a
circle a. On the circle a, two sequences
of points are given: AI' A 2 , A a
and
Bt t B2, B 8
following in the same direction and such that the straight lines
AlAI' AlAs, A aA 4
• and BIB,., BIB a,
BsB, ... touch the circle p. Prove that
the straight lines AIB I , A 2 B 2 , AaBa
touch one and the same circle whose centre
is found on the straight line passing through
the centres of the circles a and fl.

145


319. Using the result of the preceding
problem, prove the following statement
(Poncelet's* theorem). If there is one n-gon
inscribed in a circle a and circumscribed
about another circle p, then there are infinitely many n-gons inscribed in the circle
ex and circumscribed about the circle ~ and
any point of the circle can be taken as one
of the vertices of such an n-gon.
320. On the sides of a regular triangle
PQR as bases, isosceles triangles PXQ,
QYR, and RZP are constructed externally
so
that LPXQ =
(n
2 LA),
t
1
LQYR = T (n
2 LB), RZP = 3 X
(n + 2 LC), where A, B, C are the angles of a certain triangle ABC. Let A 0 denote
the intersection point of the straight lines
ZP and YQ, B o the point of intersection
of the lines XQ and ZR, and Co the point
of intersection of YR and XP. Prove that
the angles of the triangle A oBoC 0 are congruent to the corresponding angles of the
triangle ABC.
Using the obtained result, prove the following Morley's·. theorem: if the angles
of an arbitrary triangle are divided into
three equal parts each (or trisected, hence,

+-

+

+

• Poncelet, Jean Victor (t 788-1867). A French
geometer and engineer.
•• Morley, Frank (1860-1937). An English mathematician.
to-01557

t46


the relevant lines are called trisectrices),
then the three points which are the intersection points of the pairs of trisectrices adjacent to the corresponding sides of the
triangle are the vertices of a regular triangle.
321. We arrange the vertices of a triangle ABC in positive order (anticlockwise).

»<:

For any two rays a and P the symbol (a, p)
denotes the angle through which the ray
a must be rotated anticlockwise to be
brought into coincidence with the ray p.
Let a l and a~ denote two rays emanating

»<.

=

from A for which (AB, a l )

(0 C) = ~ LA, a
»<:

which (AR, (

~ (LA

2)

+ 2n),

=

2

»<.

(ai' ai)

and a; the rays for

-<.

(a 2 , a~)

=

and, finally,

»<.

»<:

(a~, AC) =

a 3 and a~
~

the rays for which (AB, aa) = (aa,
(DC)

= ~ (LA

=

+ 4n)

a~) =

(ail ai, where

i = 1, 2, 3, are called trisectrices of the
first, second and third types). In similar
fashion, for the vertices Band C we determine Pl' Pi and Yilt Yi U, k = 1, 2, 3). We
denote by aiPlYIt the triangle formed by
respectively intersecting lines (not rays)
a, and Pi, pj and yit, Yit and ai. Prove that

See. 2. Selected Problems

1.47

+

for all i, t, k such that i
j + k - 1 is
not multiple of three, the triangles ai~l'k
are regular, their corresponding sides are
parallel, and the vertices lie on nine straight
lines, six on each line (Morley's complete
theorem).
Geometrical Inequalities.
Problems on Extrema
322. At the beginning of the nineteenth
century, the Italian geometer Malfatti*
suggested the following problem: from a
given triangle, cut out three circles such
that the sum of their areas is the greatest.
In later investigations, M alfatti's circles
were understood as three circles touching
pairwise each other, each of which also
touches two sides of the given triangle.
Prove that for a regular triangle Malfatti's
circles yield no solution of the original
problem. (Only in the middle of this century was it proved for any triangle that
Malfatti's circles yield no solution of the
original problem.)
323. Prove that p~
V6Rr, where p
is the semiperimeter, rand R are the radii
of the inscribed and circumscribed circles
of a triangle, respectively.

+

• Malfatti, Giovanni Francesco Giuseppe
(t731-1807). An Italian mathematician.
10*

148


324. Prove that the perimeter of the
triangle whose vertices are the feet of the
altitudes of a given acute triangle does not
exceed the semi perimeter of the given triangle.
325. Prove that if the triangle formed
by the medians of another triangle is obtuse, then the smallest angle of the former
triangle is less than 45°
326. Let ABeD be a convex quadrilateral. Prove that at least one of the four
angles BAC, DBC, ACD, BDA does not
exceed n/4.
327. Prove that the median drawn to
the largest side of a triangle forms with the
sides enclosing this median angles each of
which is not less than half the smallest angle
of the triangle.
328. Prove that if in a triangle ABC the
angle B is obtuse and I AB I = 1 AC 1/2,
then LC > LA/2.
329. Prove that the circle circumscribed
about a triangle cannot pass through the
centre of an escribed circle.
330. In a triangle, a median, a bisector,
and an altitude emanate from the vertex A.
Given the angle A, find out which of the
angles is greater: between the median and
bisector or between the bisector and the
altitude.
331. Prove that if the medians drawn
from the vertices Band C of a triangle

t49

ABC are mutually perpendicular, then
cot B + cot C~ 2/3.
332. Given a triangle ABC, I AB 1 LBCM.
333. Two tangents AB and AC are
drawn to a circle from an exterior point A;
the midpoints D and E of the tangents are
joined by the straight line DE. Prove that
this line does not in tersec t the circle.
334. Prove that if a straight line does
not intersect a circle, then for any two
points of the line the distance between them
is enclosed between the sum and difference
of the lengths of the tangents drawn from
these points to the circle. Prove the converse: if for some two points on the straight
line the assertion is not fulfilled, then the
line intersects the circle.
335. In a triangle ABC, the angles are
related by the inequality 3 LA - LC <
n, The angle B is divided. into four equal
parts by the straight lines intersecting the
side AC. Prove that the third of the line
segments (counting from the vertex A) into
which the side. AC is divided is less than
lAC 1/4.
336. Let a, b, c, d be successive sides of
a quadrilateral. Prove that if S is its area,
then S ~ (ac
bd)/2, an equality occurring only for an inscribed quadrilateral
whose diagonals are mutually perpendicular.

+

150


337. Prove that if the lengths of the
angle bisectors of a triangle are less than 1,
then its area is less than V3/3.
338. Prove that a triangle is either acute,
or right, or obtuse accordingly as the
expression a2 + b2 + c2 - 8R2 is, respectively, either positive, or zero, or negative
(a, b, c the sides of the triangle, R the radius of the circumscribed circle).
339. Prove that a triangle is either acute,
or right, or obtuse accordingly as its semiperimeter is, respectively, either greater
than, or equal to, or less than the sum
of the diameter of the circumscribed circle
and the radius of the inscribed circle.
340. Prove that if the lengths of the sides
of a triangle are related by the inequality
a2 + b2 > 5c2 , then c is the smallest side.
341. In a triangle ABC, LA < LB <
LC, I is the centre of the inscribed circle,
o the centre of the circumscribed circle, and
H the intersection point of the altitudes.
Prove that I lies inside the triangle BOH.
342. The triangles ABC and AMC are
arranged so that MC intersects AB at a
point 0, and I AM I + 1MC I = I AB I +
I BC I. Prove that if I AB I = 1Be I,
then I OB I > I OM I·
343. In a triangle ABC, a point M lies
on the side BC. Prove that (I AM I lAC D I Be I ~ (I AB I - I AC I) I Me I·
344. Let a, b, C be the sides of a triangle
ABC, and M an arbitrary point in the


151

plane. Find the minimum of the sum
2
2
2
1 MA 1
1 MB 1 + I MC 1 •
345. The sides of an angle equal to ex
form the sides of a billiards. What maximum number of reflections from the sides
can be done by a ball (the ball is assumed
to be dimensionless)?
346. Four villages are situated at the
vertices of a square of side 2 km. The villages are connected by roads so that each
village is joined to any other. Is it possible
for the total length of the roads to be less
than 5.5 km?
347. A point A lies between two parallel
lines at distances a and b from them. This
point serves as a vertex of the angle ex for
all possible triangles, two other vertices
of which lying on the given straight lines
(one on either line). Find the area of the
least triangle.
348. In a circle of radius R centred at
0, AB is its diameter, a point M is on
the radius OA such that I AM I I MO I =
k. An arbitrary chord CD is drawn through
the point M. What is the maximal area of
the quadrilateral ABCD?
349. Given an angle with vertex A and
two points M and N inside this angle.
Drawn through M is a straight line intersecting the sides of the angle at points B
and C. Prove that for the area of the quadrilateral ABNe to be minimal, it is necessary and sufficient that the straight line

+

{52


Be intersects AN at a point P such that
I BP r = I MC I. Give the method of construction of this line.
350. The vertex of an angle ex is found at
a point 0, A is a fixed point inside the
angle. On the sides of the angle, points M
and N are taken such that LMAN =
P (a + p < n). Prove that if I AM I =
I AN I, then the area of the quadrilateral
OMAN reaches its maximum (of all possible quadrilaterals resulting from change
in M and N).
351. Bearing in mind the result of the
preceding problem, solve the following.
A point A is taken inside an angle with
vertex O. The straight line OA forms angles
cp and", with the sides of the angle. On
the sides of the former angle, find points M
and N such that LMAN = P (q>
1P
~ < n) and the area of the quadrilateral
OMAN is maximal.
352. Given a triangle OBC (LBOC = a).
For each point A on the side Be we define
points M and N on OB and OC, respectively,
~ < n) and the
so that LMAN = P (a
area of the quadrilateral OMAN is maximal.
Prove that this maximal area reaches its
minimum for such points A, M, and N
for which I MA I = I AN I, and the straight
line M N is parallel to Be. (Such points
exist if the angles Band C of the triangle

+ +

+

ABC do not exceed

~

+ ~ .)


153

353. Let ABCD be an inscribed quadrilateral. The diagonal AC is equal to a and
forms angles ex and p with the sides AB and
AD, respectively. Prove that the magnitude
of the area of the quadrilateral lies between
a 2 sin (a + p) sin p
d a 2 sin (a + p) sin a
2 sin a
an
2 sin P

•

354. Given an angle a with vertex at
a point 0 and a point A inside the angle.
Consider all quadrilaterals OMAN with
vertices M and N on the sides of the angle
and such that LMAN = P (ex,
p > n),
Prove that if among these quadrilaterals
there is a convex one such that 1 MA 1=
I AN I, then it has the least area among
all the quadrilaterals under consideration.
355. Consider a point A inside an angle
with vertex 0, OA forming angles q> and '"
with the sides of the given angle. On the
sides of the angle, find points M and N
such that LMAN = P (q>
~ > '1)
with minimal area of the quadrilateral
OMAN.
356. Given a triangle OBC, LBOC = a.
For any point A on the side Be we define
points M and N on DB and DC, respectively,
so that LMAN = p, and the area of the
quadrilateral OMAN is minimal. Prove
that this minimal area is a maximum for
such points A, M, and N for which 1 MA )=
I AN 1 and the straight line M N is parallel to Be. (If there is no such a point A,
then the maximum is reached at the end

+

+ '" +

i54


of the side Be for a degenerate quadrilateral.)
357. Find the radius of the largest circle
which can be overlapped by three circles
of radius R. Solve the problem in the general case when the radii are n; R 2 , n;
358. Is it possible to cover a square 5/4
on a side with three unit squares?
359. What is the greatest area of an
equilateral triangle which can be covered
with three equilateral triangles of side 1?
360. In a triangle ABC, on the sides
A C and BC, points M and N are taken,
respectively, and a point L on the line
segment MN. Let the areas of the triangles
ABC, AML, and BNL be equal to S, P,
and Q, respectively. Prove that S ~

3{p

+ YQ.

r

36t. Let a, b, c, 8 denote, respectively,
the sides and area of a triangle, and a,
p, 'V the angles of another triangle. Prove
that a2 cot ex + b2 cot P + c2 cot y~ 48,
an equality occurring only in the case
when the triangles are similar.
362. Prove the inequality a 2 + b2 +
2
c ~ 48 V3 + (a - b)2 + (b - C)2 +
(c - a)2, where a, b, c, S are the sides
and area of the triangle, respectively (the
Einsler-Hadoiger inequality).

363. Given a triangle with sides a, b, and

c. Determine the area of the greatest regular triangle circumscribed about the given


i55

triangle and the area of the smallest regular
triangle inscribed in it.
364. Let M be an arbitrary point inside
a triangle ABC. A straight line AM intersects the circle circumscribed about the
triangle ABC at a point At. Prove that
(BMI·leMI
.
J AIM I
~ 2r, where r IS
the radius of the inscribed circle, .an equality
being obtained when M coincides with the
centre of the inscribed circle.
365. Let M be an arbitrary point
Prove that
inside a triangle ABC.
lAM I sin LBMC + I BM I sin LAMC +
I CM I sin LAMB~ p (p the semiperimeter of the triangle ABC), an equality occurring when M coincides with the centre of
the inscribed circle.
366. Let hI' h 2 , h s be the altitudes of
a triangle ABC, and u, v, w the distances
to the corresponding sides from a point M
situated inside the triangle ABC. Prove
the following inequalities:
(a) ..!!!. + ~ +..!!L ~ 9·
u
v
W :::::" ,
(b) hth2h3~27uvw;

(c) (hi -u) (h 2 - v) (h 3 -

w)~8uvw.

367. Let h be the greatest altitude of a
non-obtuse triangle and Rand r the radii
of the circumscribed and inscribed circles,

156


+

respectively. Prove that R
r~ h (the
Herdesh theorem).
368. Prove that the radius of the circle
circumscribed about the triangle formed
by the medians of an acute triangle is greater than 5/6 of the radius of the circle circumscribed about the original triangle.
369. Prove that the sum of the squares
of the distances from an arbitrary point
in the plane to the sides of a triangle takes
on the least value for such a point inside
the triangle whose distances to the corresponding sides are proportional to these
sides. Prove also that this point is the intersection point of the symedians of the given
triangle iLemuan's point).
370. Given 8 triangle each angle of which
is less than 1200 Prove that the sum of
the distances from an arbitrary point inside
it to the vertices of this triangle takes
on the least value if each side of the triangle
can be observed at an angle of 1200 (Torricelli's point).
371. Prove that among all triangles
inscribed in a given acute triangle the one
whose vertices are the feet of the altitudes
of the given triangle has the smallest perimeter.
from a point inside a triangle to its vertices
is not less than 6r, where r is the radius
of the inscribed circle.
373. For an arbitrary triangle, prove the

+

157

·
Iit
be cos A
< p < bc+al '
mequa 1 y
b+ e
a
a
where a, b, and c are the sides of the triangle
and pits semiperimeter.
374. Let K denote the intersection
point of the diagonals of a convex quadrilateral ABCD, L a point on the side AD,
N a point on the side Be, M a point on the
diagonal AC, KL and MN being parallel
to AR, LM parallel to DC. Prove that
KLMN is a parallelogram and its area is
less than 8/27 of the area of the quadrilateral ABCD (Hattori's theorem).
375. Two triangles have a common side.
Prove that the distance between the
centres of the circles inscribed in them is
less than the distance between their noncoincident vertices (Zalgaller's problem).
376. Given a triangle ABC whose angles
are equal to a, p, and y. A triangle DEF
is circumscribed about the triangle ABC
so that the vertices A, B, and C are found
on the sides EF, FD, and DE, respectively,
and LECA = LDBC = LFAB = <p. Determine the value of the angle q> for which
the area of the triangle EFD reaches its
maximum.
377. In a triangle ABC, points At, B l ,
CI are taken on its sides BC, CA, and AB,
respectively. Prove that the area of the
triangle AIBtC l is no less than the area of
at least one of the three triangles: ABlel ,
AtBCI , AlBIC.


i58

378. Let 0, I, and H denote the centres
of the circumscribed and inscribed circles
of a triangle and the point of intersection
of its altitudes, respectively. Prove that
I OB I~ IIH I V2.
379. Let M be an arbitrary point inside
a triangle ABC; x, Y, and z the distances
from the point M to the vertices A, B,
and C; u, v, and w the distances from the
point M to the sides BC, CA, and AB,
respectively; a, b, and c the sides of the
triangle ABC; 8 its area; Rand r are the
radii of the circumscribed and inscribed
circles, respectively. Prove the following
inequalities:
(8) ax
by
cz~ 48;
(b) x + y
z~ 2 (u
v
w)
(Herdesh's inequality);
(c) xu + yv + zw~ 2 (uv
vw + wu);

+

+
+

+ +
+

1
1)
1
1
1
-+-+- &-+- .t--.
y
z
v
w'
R
(e) xyz~2T (u + v) (v + w) (w + u);
1
(d) 2 ( x

(f)

~u

4R

xyz~-

r

uvw;

(g) xy+yz+zx~ 2R (uv+vw+wu).
r

380. In a given triangle, we draw the
median to the greatest side. This median
separates the triangle into two parts. In
each of the triangles thus obtained, we

Sec. 2. Selecte d Problems

t59

also draw the median to the greatest side,
and so forth. Prove that all the triangles
thus constructed can he divided into a
finite number of classes in such a manner
that all the triangles belonging to the
same class are similar. Also, prove that
any angle of any newly obtained triangle
is no less than half the smallest angle of
the original triangle.
381. Find the triangle of the least area
which can cover any triangle with sides
not exceeding 1.

Answers, Hints, Solutions

Section 1
17. The angle bisector separates the given
.
triang1·
e Into two parts W hose areas are al sin 2 '
2" · a

~sin ~
ab .
S1n a ;

T

, and the area of the entire triangle is
al
hence ( T

+T ' Sin '2 ==- T Sin a,
bl) . a
ab.

a

2abcosT
l

=

a+b
t 9. Let us take a circle touching the sides A B ,
BC, and CA. If this circle does not touch the side
DA, then drawing the tangent DAJ.. (Allies on AB),
we obtain a triangle D A A 1 in Which one side is
equal to the sum of the two others.
20. Drawing through the vertices of the triangle
straight lines parallel to the opposite sides, we get
a triangle for which the a1 ti tudes of the original
triangle are perpendiculars to the sides at their
midpoints.

21

•

a+b
2

22!.. .. /3
• 2 V

va
:t'

23.

Y2-1 (a+b- ../a2+b:.
"-)
2
r

25.

eta.

28. I

a-;b I

29.

m2

va

24. - 2 - '

~(a-b)lsina..

t6i

30

In-a.

h

3t

·2 tan - , - .

•

300

·

32

ab

• T·

33. 90°. 36. r l (2 V3+3). 37. I

Va (21-a).

t

38. 2(8 1+81) ,
39. If a > b, then the bisector intersects the
lateral side CD; if a < b, then the base is BC.
2ab
t-k
40. a+b.
41. arccos t+k.

a+b ,r
42. - 4 - J' 3b'+2ab-al •
44·

i .. /8
T .V 2·

45.

(VS1+VSJ3.

43. al.

41. I

=+:

46. 90°+ ~ •

I Vall+b ll• 48. arcsin

(+-1).

49. (6-n): 2n: (6-n).

SO. ~ (V2-t) [(2 V2-1)n-4).

a
T
l

5t.
53•

(6

.r y

3-6-n).

..!.. ..t'rbl-al •
2

58. If a. < 90°,

~

54.

<

52.

J!.. •
3

RI
T (n Va) ·
3+2
55•

..!-S'
9

90°, then the angles of
(z, 90°- ~, CI +
if
(I> 90°, ~ < 90°, then they are a-OOO, 90 +P,
tBO° - a - ~; if CI < 90°, P> 90°, then they are
iOo+e.t, ~-goo, iSOo-a-p.
t .. r - a
.
36
59. T f ml - 4S . 60. 5 . 6"li- 25 hi,
6. ABC are equal to 90°-

62.

V

11-01557

n

(4n~-1) •

p;

162


63. In an isosceles triangle with the vertex angle
of n/5, the bisector of the base angle separates the
triangle into two isosceles triangles one of which is
similar to the original one.
Answer:
1i4. R2

[cot

~

R.

-+

(n-a.)

a (4 Si~2 a+ t)
B SiD a
·

66.

68.

V5-1
2

a2+4r2
4r

J.

3a

69.
3

(a.
tan2"-cota ) .

ex-~

a

79.

t

V7

va'

77.

R"-al
•

75.

(Va 1.)

aT-+-"'2 ·

T(P+y-cx).

y 10.

70. a ViO
4

2(5+V13)'

a cos
2
74.
sin (a+~)

:

67. 2r2 (2 V3-f-3).

a
71.2.72. 4(a a b 2 ) ' 73. 2
2+b

76. 3

65.

80.

ac+bd
a

2R

78.

•

al

V3

12'

•

81. 2 . 2 n . 2P • 82. 1 b~a IV 4dZ - (b - a)2.
sin ex SlD
83. 2 (R2+a t ) .
84. Two cases are possible: the two centres
are on both sides of the common chord and on
the same side of it. Accordingly, we have two

pairs of answers: a (V3-1). a 1~2 (va -1) and
a

V2(-./'-) ·
( ..yr« 1. ) , a 2
3+
J' 3+1

86.

3- V7
4
.

i63


87. V13. 88. arccos

+

89.

I

90.

t ± V1-2k
. 2

~I.

~

91.

I

,

·

I

tt+

~- ~

I

:Tt
P :Tt
--- a.-t---92. al 2 V3-3 (Generally
2
2
2·
8·
speaking, two triangles are possible, but in one
of them two vertices lie on the extensions of the
diagonals.)
7 V2
br
..r
93. -We 94.
95. y 7.

c.

R y 3-1.
96. -2 ( ..r - )

100.

+V

..rVi
97." 10. 98. - - - t.
cos a.

va.

sin ~

t a+~

102

96-54

• a sin a co

103.
105.

t

2

lOt. 3: 4.
•

~

ill V 25al+c'+10ac cos p.

107•

al+bl-2ab cos a.

4 VRr(R-r)
f06
6Rr-rl - RI ·
·

.. /' bl
:0 c.

108.

V

2 (b-acosa)

+a

l+2ab

ff2.
113.

s cosSa..

1tO. V4RI-a l •

V+ + +
bl

n

4

1-

;

~
2

T
b
ru. T.

l/ al+b2+2ab cos a·1 cot a I·
abcostt.

n.

1t4 • arcsin 2 an d n - arcsin 2
·
·
11*

sin

tt

2 cos

tOO.

3
104. , S .
•

164
tt5.
.

a2 (V2-

i). 116. tJ cos (a.+~)
cos (2a.+ 1')

,

a sin

(a.+~)

COB (2a

+ 6> •

a.
1
2 cos 3'+3
117. T a (b-a cos a.) sinS a. 118.
a..
6cosT+ 1

119•

2 Y 8. (8 1 +8 1)

---...~;:;;===;::;;--

V4S!-SI

120. 4 cos]-

12t.

V

1~.

(Rt-R 1 ) (R. Sinll.~+Rl cost]) .

~I +

122. {

sinl!. COSI a.+1'

2
a l cost

IX

2

•

-

2
123. Val+bl-ab, Val+bl+ab. t25. 15°, 75°.
R Vi
. /~r126. - 8 - . 127. 2 , 6. 128. f 2.

129. !(2 Y3+3).
3

13t.
R

130. 2RI. sinS a sin ~
SiD

3

va ~J3 -1).

«(I + 1') •

132. 1.1

133. If 4/4

<

< a/2 there is only one solution: al / (t 6R ).
< R ~ 4/4 or R ~ 2, we have two solutions:
t

If 0

a'/(16 R) and a'/(8 R).
n

134.

136.

T

RI_rl

and arccos RI+rl

aY7/4.

•

135. 30°.

137. R(3-2V2)/3.

.. lOt-cos ~
39
ab tan a.
138. 4 V a-cos ~. 1 · Val tan' cz+(a-bl) •
(In the triangle ONP, KP and N M are altitudes,
therefore OA is an altitude.)
140.
2Rr/(R +r).
141. a/2.

165

143. The error does not exceed 0.00005 of the
radius of the circle.

t". 113-56

V3.

1.7
1•.

11.Q

]/3+ Vt5

t46. 3 1~

t45. 7.5.

150.

231
3'
4

~.

~rt'

3. 151.

2

•

2

V3 •
3

152.

V5
T'

11.9

~.

16
-)
9 ( . , v 7.
4-

t53. 2r2 sin 2 a. sin 2a.. tM. 2 -}.
155.

5 t
12 n+T arccos (3 V3) ·
1t-T

156. Vi'2(2158. If a <

va).

157. ar/(a+2r).

1t

'3' then the problem has two

solutions: R2 sin a. ( 1 ± sin
the only one: R2 sin a. ( 1

~

+sin

~ ~ a. <

) ; if

31,

~ )•

159. From ; (3 V2-4) to ; .

160.

From

I al-bl I

2abc
a'+b l to 1. 161. ab+bc+ca'
(Through an arbitrary point inside the triangle,
we draw three straight lines parallel to its sides.
Let the first line cut off the triangle which is sim ilar
to the original one with the ratio of similitude
equal to A, the second line, with the ratio equal to
fit and the third-with y. Prove that A JJ.
y = 2.)
Rr
f62. . R+r •

+ +

163: Take on the line BA a point Al such that
A, A. D and C

I AlB I = I Ale I. The points

tOO


lie on acircle(LDA 1C=90o - LABC=LDAC).
Consequently,
L.AtAC = L.AtDC = 90°, and
hence LBAC = 90°.
t
13
164. 1. 165. 2 T. 166. 15 a.

a'+a Va2+8bS

167.
4
169. 6. 170. '3.
171. If Q;;;;'

va (.r- 3 vS

168.

a2-+-a (db-b) Ybd.
a-

+

S, then the desired distance is

..yr-) · And if Q < T1 S,
Q

answers are possible:

f72. 3rt

•

II-k'i
1 +k
S

then

two

~3 (Vs ± VQ).
•

t73.

2(I+COS;)
1+sin ;

(a'+ bl - cS) c
400
•
175. Let A and B denote two adjacent vertices
of the rhombus, M the point of intersection of its
diagonals, 0 1 and 0 1 the centres of the circles
2
(0 1 on AM, 0, on BM). We have: lAB 1 =
4t

174.

bAO~I~, ~ I ,Bt:J'I') J I Zr4+I~, -=- bOJ~'~l
I OtM 12 ) = R2 + ,2 - al.
Answer: YRI+ r2 - a 2 ,

+

8Ra,.a
176. (R2+ r l)l .
177. I AB I

V al + bl + 2ab cos C£

if B lies
sin a.
inside the given angle or inside the angle vertical

167

V al +bl-2ab cos a in the reto it; I AB 1==
sin a
maining cases.
3 113
.
178· 2arcsin l (hhahbhb)·
179. 51t- 3 ·
a+
180. Since EF is perpendicular to CO (0 the
point of intersection of the diagonals), and the
conditions of the problem imply that A C is the
bisector of the angle A which is equal to 60°, we
have: 1AE 1 = I AF I = I EF I. If K is the midpoint of EFt

a~3,

V3 co I =

then I AO 1= 2a 3 ' I

ICKI.IOK=IEKI 2 =
al

va

Answer: - 4 - and 2al

181.

3
'4 h.

+IAKI2.

,.rJ' 3.

182. Denote: L BAC = L BDC = a,

L CBA= L BCD=P, L BAM=(J). Then
IBM 1+1 MC I
sinq>+sin(a-q»
1 AM I I MD t = sin(~+a-q»+sin (P+q»

+

(a q> )
2'a) 2
(~+ 2 cos (a. - q> )

• ex
sin "2 cos

sin

sin ex.

sin <p+a)+sin p=

c

a+b •
183. There is always a chord parallel to the
base of the triangle. The chord is divided by the
lateral sides into three equal parts (undoubtedly,

0< a < 2). Its length is 2a2~1

In addition, if

< 1/ V2 t then there exists one more chord, which
is not parallel to the base and possesses the same
property. The length of this chord is 3/ V 9 - 2a2 •

a

t68

184. Let Be and AC intersect MN at points P
t·
·
an d Q ,respective1 S e ting I MC I, z; we
y.
,eN

have:

IMPI

IPNI

SBMC
SBNC

IMBI·IMCI.

IBNI·ICN,

3x
Hence, I MP 1= 3x+4. Analogously,
x

.1:+ 1

3%

=4-

I MQ 1=

• For x we get the equation

x

x+ 1 -

3%
3%+4 =a, 3az2+(7a - 1)%+4a=0. Since
D :.> 0 and 0 < a < 1, the greatest value of a
is equal to 7 - 4
185. The equality SABN = SCDM implies that
5MBN = SMCN since MN is a median of the
triangles ABN and CDM. Hence BC II MN and
AD II MN, that is, ABCD is a trapezoid with bases
AD and BC.

va.

5k- 2 ± 2 V2k (2k-1)
2-3k
186. We have: I AD I :>- I DM I - I AM I

Answer:

2. On the other hand, I AD I

~

=

IBDI

-:-"600 = 2.

81n

Consequently, I AD I = 2, AD is the larger base,
and th.e point M lies on the line AD.
Answer:

Vr:

187. Let BD denote an angle bisector in a
triangle ABC, At and C1 the midpoints of the sides
Be and AB, I DA 1 I = I DC I I. Two cases are
possible: (1) LBA1D = LBC1D and (2) LBA1D
LBC1D = 180°. In the first case lAB 1 = I BC I.
In the second case, we rotate the triangle ACID
about D through the angle C1DA 1 to carry Ct

+

into A 1 - We get a triangle with sides

(1.~C

{G9

a

+e

be

.

- 2 - , a + e (a, b, and c the sides of ~ABC),
which is similar to the triangle ABC. Consequently,
ba
a +c
be
-+ : a = - 2 - b = -+ : c,
a
c a e

so a + e

==

b V'2. Since a =1= c, at least one of the two
inequalities b =1= a, b =1= c is true. Let b -=1= c, then
b + e = ,. V2, b = a, and we get a triangle
with sides a, 4, a (V2 - I), possessing this
property. Thus, there are two classes of triangles
satisfying the conditions of the problem: regular
triangles and triangles similar to that with sides
i, 1, V2 - t.
188. If a is the angle between the sides a and b,
then we have: a + b sin a ~ b+a. sin a, (a-b)X
(sin a; - 1) :>- 1, sin a> t. Hence, ex, = 90°.
Amwer: Va 2 + v.
189. Prove that of all the quadrilaterals circumscribed about the given circle, square has the
least area. (For instance, we may take advantage
of the inequality tan ex,
tan p:>- 2 tan (ex, + P)/2]
where a; and p are acute angles.) On the other hand,

+

SABCD ~

t
2(1 MA 1·1 MB I + 1MB 1·1 Me I +

I MC I" MD 1+' MD
I MB II) +
I MD

i-

(I MB

II) + ~ <I

r I MA

I)

:s;;~ (IMAI2+

1
1 + I MC 11)+}<lMCII+

MDI 2

+

2
I MA 1 )

=

1.

Conse-

quently, A CBD is a square whose area is 1.
tOO. Let us denote: I BM I = z, I DM I = y,
I AM 1= t L.AMB = cp. Suppose that M lies
on the line segment BD. Writing the law of cosines
for the triangles AMB and AMD and eliminating
cos q>, we get: I' (z + y)
zy (z + U) = a2 y
cJIz. Analogously, we get the relationship zt (z+y)+

+

+


170
zy (z

(c2

-

+

y) = b2 y
d2) s;

AnMer:

+

c2z.

Thus,

(a2

-

bl ) y

=

I;:_~: I·

t 91. If the vertices of the rectangle lie on the
concentric circles (two opposite vertices on the
circles of radii R 1 and R , and the other two on
the circles of radii R 3 an~ R,), then the equality
Rl RI = RI Rl must be fulfilled. Let us
prove this. Let A denote the centre of the circles,
the vertices K and M of the rectangle K-LMN lie
on the circles of radii R 1 and R t , respectively, and
Land N on the circles of radii R a andR" respectively. In the triangles AKM and ALN, the medians
emanating from the vertex A are equal, the sides
KM and LN are also equal. This means that our
statement is true.
Let the second side of the rectangle be %, z > t.
The radii R t , R t , R a, R 4 are equal, in some order,

+

+

-

+-

t

~r

to the numbers I , z; 'f Zl
1, 2 y zI+ t.
Checking various possibilities of the order, we
find: x2 = 7, R l = it R 1 = 2 Y2, R a = V2,
R, = ,,1"7.
Consider the square KtLtMlN t with side y
whose vertices lie on the circles of radii R 1 = t,
R a = V2,
R , = 2 Y2, R 4 = V7. Denote:
L.AK1L 1 = cp, then L.AKIN l = 90° ± cp or cp ±
90°. Writing the law of cosines for the triangles
AKIL I and A KtN t, we get
1+ZI_2zcosq>=2,
{2zcoSq>=XI - 1 ,
{ 1+z2 ± 2% sin q>=7, => ± 2% sin q>=z2-6.
Squaring the last two equalities and adding the
results, we get: 2z& - 10%2
37 = 0, zl = 5 ±

t

+

-

2" V26.
AnBIDer:

V5 ± 2 Y26.

.


t71

192. Let us first prove the following statement.
If the perpendiculars to AB and BC at their midpoints intersect A C at points M and N so that
I MN I = ). I AC I, then either tan A tan C =
1 - 2)" or tan A tan C = 1 + 2).. Let us denote:
I AB I = e, I BC I = a, I AC I = b. If the segments of the perpendiculars from the midpoints of
the sides to the points M and N do not intersect,
then

I M N I = b-

c
2 cos A

A) sin B cos A cos C =

a
2 cos C

+

(sin 2C

)"b--'-2(t
--r-

+ sin 2A)

-

=>

2 (1 - ).,) sin (A + C) cos A cos C =
sin (A
C) cos (A - C)~2(t-A)cosA X
cos C = cos A cos C
sin A sin C
tan A X
cos C = i - 2).,.
And if these segments intersect, then tan A tan C=
t
2).,. In our case A, = t, that is, either tan A X
tan C = -1 or tan A tan C = 3. For the angles
Band C we get (A = 1/2) either tan B tan C = 0
(this is impossible) or tan B tan C = 2. The
system
tan A tan C = - 1,
tan B tan C = 2,
{

+

+

*

+

A+B+C=n

has no solution. Hence, tan A tan C = 3. Solving
the corresponding system, we find: tan A = 3,
tan B = 2, tan C = 1.
An8w~r: n/4.
t 93. Let R denote the radius of the circle circumscribed about~ABC, 0 its centre, N the median
point of the triangle BCM. The perpendicularity
of ON and cu implies the equality I cs I' I··MN 12 = I CO 12 - I OM II. Let I AB I = it
I MB I = x,
I cu I = y, then I MN 12 =
t
i
2 - k 2 ) , I cs II = 9 (2y 2 + 2k 2 - x2) ,
T(~y2
2x

+

172


I CO I' = Rt,

10M II = RI cos' C

+ (x- ~)~

We get equation for z: 2x" - 3%

+ k" = O.

(if

1 < Ie < 3

Answer:

3±

":9=8fs

r

2

I

then both points are found inside the line segment AB).
194. If 0 is the midpoint of AC, then 1 AB 11 =
1
1
, BO 1 I AO I" := I BK II - I KO 1 lAO 11 =
I BK II
1AO I - I AK 1)( 1AO I
I AK 1)=
1+1 AK 1·1 CK I=bl+bd.
I BK 1
AnSIDer: V hi bd •
195. (1) The length of a broken line of three
segments is equal to the line segment joining its
end points. This is possible only if all of its 'vertices
2ab
2ab
"
IIe on t h·ISsegmen t · x =
• r- , y = . r
•
a+bv3
av3+b
(2) x, y, I are the sides of a triangle whose altitudes area, b, and c. Such a triangle must not be
obtuse-angled. To find x, y, I, let us take advantage of the fact that a triangle whose sides are
inversely proportional to the altitudes of the-given
triangle is similar to the latter.
1
t
t
z=-2-' Y=-2b ' %=-2 ,where
as
8
cs

+
+(

+
+

+

'=VP{p- ~) (p-~) (p- ~). 2P=++

i+ ~ · Tbe problem bas a solution if :, + :1 ~
t

1

1~1

t

t

t

Ci" ' bI+CI~7 ' -cs+ar;;'bI·

(3) Consider the points A (a, b), B (or, 0),
C (0, y) in a rectangular coordinate system. It

follows from the given system that ABC is an
equilateral triangle. When rotated about A through
an angle of 60° in the appropriate direction, the


f73

point B goes into e. We can find the equation of
the straight line into which the x-axis is carried
by this rotation. (In particular, the slope is equal
to ± V3:)
A nnaer: z = - 4 ± b
1/ = -b ± 4
(4) If z:>- 0, y > 0, I :>- 0, then e, y, J are the
distances to the vertices of a right triangle ABC
in which the legs Be and CA are a and b, respectively, from such a point M inside it from which
all of its sides can be observed at an angle of 1200 •
To determine the sum z
y
I, let us rotate
the triangle CMA about e through an angle of 60°
in the direction external with respect to the triangle ABC. As a result, M and A go into M1and AJ.t
respectively. Then BMM1A 1 is a straight line and,
consequently, z
y
z = I BM I
I CM I +
I AM I = I BA I I =
a l + b2
ab
Analogously, we consider the case when one of the
variables is negative (generally speaking, not any
of them can be negative) and other cases.

va,

V3.

+ +

+ +

Y

Va' +b

Answer: ±

+

± ab

l

+

va.

va.

t96. Let z be the distance from the centre of the
square to the straight line I, q> the acute angle
formed by one of the diagonals of the square and
the line I. The distances from the vertices of the
square to l are equal to (in the order of traverse):

V2 SID cp,
.

V2 cos cp,

z+a 2

z+a 2

I

I, I

z-a

~2 sill cp

pothesis,

I

zl-

z -

~I

Silll cp

a

~2 cos cp

I= I

zl -

I.
~I

By hy-

cost cp

I'

whence either tan' '1'= 1, which is impossible,
or zl=o,l /4.

Answer: a/2.

197. From the condition LB = 2 L e there
follows the relationship for the sldes of the tri-

t74

+

ac, Looking over b = 2c, a =
angle: bl = cl
2c, b = 2a, and a = 2b, we choose a == 2c since
in other cases the triangle inequality is not
fulfilled.
Answer: L C = n/6, L B = n/3, LA == n/2.
198. Let D be the midpoint of BC. We have:
b' = I BM P&=(IBD 1+ IDN I) (I BD 1- I DN I~ ==

I BD II - I DN II = I AB I' - I AD I' - I DN I ==
2
(a
b)2 - I AD 1 - I DN I'. Hence, I AN pi ==
I AD 12
I DN II = (a
b)' - b2 = a2
2ab.
Answer:
al
2ab.
199. We take on BC a point N such that the
triangle ABN is similar to the triangle ADL. Then
L NMA == L MAK
L KAD = L MAB
LDAL = LMAN.
Consequently,
I MN 1=
I AN I = k I AL I.
a

+

+

Y +

+

+

+

+

Answer: k'+b.

200. 2

V pq.

201. (a)

;,

V (R ±

z) (R ± y), the plus sign

corresponding to external tangency of the circles,
the minus sign to internal. (b) ~

Y (R +z) (R -

y).

202. Let I AM I : I MC I = k. The equality
of the radii of the circles inscribed in the triangles
ABM and BCM means that the ratio of their
areas is equal to the ratio of their perimeters.
Hence, since the ratio of the areas is k, we get
13k -12
.
IBM I =
1- k • I t follows from this equality, in particular, that 12/13 < k < 1. Writing
for the triangles ABM and BCM the laws of cosines
(with respect to the angles BMA and BMC) and
eliminating the cosines of the angles from those
equations, we get for k a quadratic equation with
roots 2/3 and 22/23. Taking into account the limitations for k, we g~t k = 22/23

175


203. Let ABC denote the given triangle, 0, X,
H the centres of the circumscribed and inscribed
circles, and the intersection point of the altitudes
of the triangle ABC, respectively. Let us take
advantage of the following fact: in an .arbitrary
triangle the bisector of any of its angles makes
equal angles both with the radius of the circumscribed circle and with the altitude emanating
from the same vertex (the proof is left to the reader).
Since the circle passing through 0, K, and H contains at least one vertex of the triangle ABC (say,
the vertex A), it follows that 1OK 1 = I KH I.
The point K is situated inside at least one of the
triangles OBB and OCH. Let it be the triangle
OBH. The angle B cannot be obtuse. In the triangles OBK and HBK, we have: 1 OK I = I HK I,
KB is a common side, L DB K = L H B K. Hence,
6.0BK = b.HBK, since otherwise LBOK
LBNK = 1800 which is impossible (K is inside
the triangle OBH). Consequently.] BH 1=1 BO 1=
R. The distance from 0 to AC equals 0.5 I BB 1=
O.5R (Problem 20 of Sec. 1), that is, LB = 60 0
(LB is acute), I AC 1= R V3. If now AI, B 1 ,
and Clare the points of tangency of the sides B C,
CA, and AB to the inscribed circle, respectively,
'then 1BA I I = I BC I 1= r
I C~l 1+IAC11=
I CBtl
t RtA I = lAC 1 = R V3. The perim(R + r). It
eter of the ttiangle is equal to 2
is now easy to find i ts area.

+

va,

+

Answer:

va

va

(R

+

r) r,

204. Let P be the projection of M on AB,
= a
e, Then I PB I = a - z, J M P I =

I API

+

1/=Val - zl ,
24-(a+z)

aV2

IANI=(a+z)
a

a Vi

a V2+y =

a

V

2+y

.INBI=

Vi (4-Z+1/ Vi)
a V2+y

Problems in Plane Geometry"

t76

I AL I =

II

V2(1I+~+" V2) •
a

V2+y

Hence
1+1
I AL 1

NB

1
1= (

1

44
,.-

aJ!2+y

4a1

21/1+%1)- ( . / -

ay2+11

)2

( 1

4

r~

( 1
4

+ 2 f.rx:
211,,+

+2 ..f r2ay+2y l + (01 -

yl») =441.
205. Let % denote the side of the triangle, and
the sides emanating from the common point of the
circles form with the straight line passing through
the centres angles ex and P; a ± P = 600 , then

:R .

i.

cos ex =
cos J =
(or vice versa). Finding
sin ex and sin P from the equation cos {ct ± p> =

~.

we determine the side of the regular triangle:

RrV3

Y RI + ,.s -

Rr ·

206. We draw a straight line BA anddenote
by D the second point of intersection with the
smaller circle. Consider the arcs AB and AD (each
less than a semicircle). Since the common tangent
to the circle at A forms equal angles with JIB and
AD, the central angles corresponding to these
I AD I
r
arcs are also equal. Consequently, 1m = R'
AD

=

a

~, I Be I =

IIYR;r.

YI BD

°

I · I BA I =

207. Let 01' 01' and
denote the centres of
the circles (the first two touching AB), %, N, and R
their radii, respectively. The common tangents to
the c~,cles centred at 0 1 and O,t O~ and 0, 0 1 and

177


o are equal to 2 V Z1/, 2 VRz, 2 VRy, respectively. By hypothesis, 2 vi; 4. Consider the
right triangle OlMOs with the right angle at the
vertex M; 0tM is parallel to BC, I 0 1 I =
z
y, 10 1 M I = 2R (z
y), I DIM I =
I 2 V Rz - 2 V Ry I (OlM being equal to the
difference between the common tangents to the
circles with centres 0, 1 and 0, 0 1 ) . Thus,
(z+y)!=(2R-z-y)I+(2YRz - 2 VBy)'.
whence R = 2 V zy = a.
208. Note that 1
4 is a parallelogram with
angles ex and n - ex (0 1 1. A C and DID' II A C.
hence, 0 10 4 11 0IOa, etc.). f K is the midpoint of
AM, L the midpoint ofMC, then I 0.0. I =I~L' =
:::3

+

°,

+

°

°°'1,0°1
,°

I AC I
2 sin ex·

Ana1ogousIy,

consequently,

8 0 10 , 0 . 0 .

=

°

8lna

I -_ 2 sin a
BD
1:1
I AC I · I BD I sin ex
,

0

4 sin l ex

SABCD

2 sin l ex •
A n'lD~r: 2 sin! a.
209. When intersecting, the angle bisectors of
the parallelogram form a rectangle whose diagonals are parallel to the sides of the parallelogram
and are equal to the difference of the sides of the
parallelogram. Consequently, if a and b are the
sides of the parallelogram and a, the angle between

them, then S
S

=

ob sin

lX,

Q=

~

(a - b)1 sin

lX,

2ab

Q ---

(a- b)1 •

.
A nllD~r:

S+Q+VQ'+2QS •
S

210. Let z denote the area of the triangle
OMN, y the area of the triangle CMN, then
12-01557

178

Probiems in Plane Geometry

ION'
%
83
I VA I =S-;=::8;'

8 1+81

8
%=-s;- , II AM I,
MC

Th e soug ht - f or

Sa+z+y.

. 8 1+

183

area

y

%

=

is equa I t 0

8 183 (8 1 +8 1 ) (8 3+8 2)
8 1 (81-8 183 )
21t . Let in the triangle ABC the angle C be
a right one, M the median point, 0 the centre
of the inscribed circle, r its radius, L B= a;

I AB I = r (cot ~ +cot ( ~ -- ~ )) =

then

r

.

sin

vi

a.
(1,)
T Sln ('"
T-T
J-

.

10M I =r,

ICOI=r It 2,

I cu I =-!. I AB I•
3

L OeM

n
= a. -4.

Writing the law of cosines for the triangle COM,

we get 1 =2+ (

8

9 2% -

z=cos

n)
(T- a ,

--)2
JI 2

whence

(

8%

y_) • where
V2 •
6

3 2% 2
4 V6-3

%=

4 V6-3 Y2
arccos
6
•
212. Let each segment of the median be equal
to 4. We denote by :z the smallest of the line
segments into which the side corresponding to the
median is divided by the point of tangency. Now,
the sides of the triangle can be expressed in terms
of a and a; The sides enclosing the median are
a y2"
z, 3a y2"
z, the third side is
2a y2"
2%. Using the formula for the length of
a median (see Problem 1 t, Sec. 1), we get 9a 2 =
1
~r"4 [2 (4 ~ry 2
x)2
2 (3a r 2 + x)2 Answer:

1t

T ±

+
+

+

+

(2a

Y2 + 2%)2],

+

whence :z = a

V2i4.


179

Answer: 10: 5 : 13.
213. Let I BC I = a, L C > LB, D and E be
the midpoints of A B and A C. The quadrilateral
EMDN is an inscribed one (since L MEN =
LMDN = 90°), I MN I = at I ED 1= a/2, MN
is a diameter of the etrcle circumscribed about
MEND. Consequently, LDME = 30°, LCAB =
90° - LEMD = 60°, LCBA = LEDN =
LEMN = LEMD/2 = 15°, LACB = 105°.
Anlwer: LA = 60°, LB = 15°, LC = 105°
or LA = 60°, LB = 105°, LC = 15°.
214. We denote by K and M the points of
intersection of the straight line EF with AD and
H

A

6

t'

D

K

a

Fig. 1
BC, respectively. Let M lie on the extension of
BC beyond the point B. If I AD I = 3a, I Be I =

a, then from the similarity of the corresponding
triangles, it follows that I DK I = I AD I = 3a,
I MB I = I Be I = a (Fig. 1, a).
In addition, I "ME I = I EF I = I FK I. If h
is the altitude of the trapezoid, then the distance
t2*

{SO


fromEtoADiSequalto: k,SBDK = ak,SBDF =
t
ah
t
"2 S EDlf = 4' = T S.
If the line EF intersects the base Be at a point
M, then I BM I =
a (Fig. i, b). In this ease
I EK 1_ .5 _ 6
.
I MK I - 2 · 3" - 5 and the distance from E

j

to AD/ equals : h.
t
6
9
-·34·- h = -8.
4

5

Anu~r:

80

that

SEFD

=

~

SEDK

=

20

1

T

S or

9

20 8 .

'{

215. Let 0 be the centre of the inscribed circle,
M the midpoint of Be, K, L and N the points of
tangency of the inscribed circle with the sides
ACt AB, and BC of the triangle, respectively. We
denote: I AK I = I AL I = e, t CK I = I CN I =
lit I BL I = I BN I = It 11
I = tI.
By hypothesis,IOMI
r. Consequently, I NM I =

=-f -

Y J OM II -

1
I ON 1 = } /

+

~I -ar and

one of the

line segments, either II or s, is equal to

.. /.1 u
V T- ,

a

;-

+ .. ;-ar--

and the other to 2' V T- a T •
Equate the expressiene for the area of the triangle
by Hero's formula and S=pr: Y(z+y+z) %11%=
ar
(z+lI+ z) r ~ zar= (Z+4) r l ~ z = - - . Thus,
4-r
the sought-for area is equal to (~+(I) r=
a-r


181

216. Let us prove that if C1 and C (Fig. 2) are
situated on the other side of Be than the vertex At
then the centre of the circle circumscribed about
the triangle CCtC. is found at the point 0 on the
side AB, and I BO 1= : 1AB I. Drawing the
altitude eM from the vertex C, we obtain the
quadrilateral BC1CM which is a rectangle. Hence

A

1l

Fig. ,2

the perpendicular drawn to ce l at its midp~nt
passes through o. Taking into consideratioythat
1
/
'CIC S II BD and I c.c, 1= '2 1BD I, w~~e that
the middle perpendicular to C1C, also passes through
O. Now, we find easily the desired radius: it is
2
equal to VI eM 12 + , MO 1 =
3a2
42
(J
.. r f 13.
4
16
4
217. Consider two cases: (t) the feet of the
perpendiculars are found on the sides of the parallelogram, and (2) one of the perpendiculars does
not intersect the side on which it is dropped. In
the first case we arrive at a contradlction, while
·
In t h e secon d case we 0 btai cos ex = a'2ab 62 t
am
l +

Y-

+-= -


t82

where a is the acute angle of the given parallelogram.
218. Expressing the angle PQN in terms of the
angles of the triangle and bearing in mind that
LPMN
LPQN == f80°, we find: LPMN ==
60°; hence LNPQ == LQMN == 30°, LPNQ ==
LPMQ == 30°, that is PQN is an isosceles triangle
with angles at the side PN of 30°, IPQI == IQNI ==

+

tlva: It follows from the conditions that ABeD
219.

is a trapezoid, BC II AD, and AC is the bisector of
the angle BAD; hence I AB I == I Be I, analogously, I Be I = I CD I. Let I AB I = IBCI =) CD 1=
a, I AD I == b. The distance between the midpoints of the diagonals is 2r, consequently b ;- a =
2r. We draw the altitude BM from the point B
on AD and we get that I AM I

=z

b-a

-2- = 2r,

I BM I = 2r. Consequently, a = I AB I == 2r
b

=

4r

+ 2r V2.

V2,

+

A nalDer: 4,-1 (V2
1).
220. Let us denote the angles A, B, and C by
a, p, and 'V, respectively. Let H be the point of
intersection of the altitudes, D the centre of the
circle passing through A , H, and c. Then L HOC =
2 LHAC = 2 (90° - 'V), LHOA = 2 LHCA =
2(90° - a). But LADe == 180 0 - 6(sinceBAOC
is an inscribed quadrilateral), 2 (900 - y)
2 (90° - a) == 1800 - p,
360° - 2a - 2'' ==
180° - p, 2P = 180° - p, P == 60°, I AC I ==
2R sin p ==

+

va.

::r~: = A, we have:

221. Denoting the ratio
SMCP

=

T

T'

SCPN=).,Q, SMCP=).,SCPN;
_

quently, (TIQ)-A

3

I AC I . I BC I
_
Me lIeN I SCMN-

_

t

conse-

SABCJ

t83


()'~i)1 (

r

+).Q)

().ti)1 (T+).IQ) =

+ 1)3Q = (TIJs +Ql/3)3.

(A

222. If 0 is the centre of the circle, then the
area of ~ OMN is ~R times the area of ~ KM N.
a-

U L MON=a., then

~I sin a. =

a!:...R S' sina.=

2aS
. C£
.r
RI(a-R) , 'MN I =2R SIn T=R , i-cos ( 1 =

V V

R

i

±

i

R'

~s:.IR)1 •

The problem has a

solution if S ~ RI (a-R)
24
223. If L BAC = L BCA = 2a., then by the law
.
2m sin 2a
of SInes, we find: 'AE I =
. 3 ,I AF I =
SIn a.
I AE I _ 2m sin 2a
Thus ~ m _ 2m sin 2a
cos a - sin 3a. cos a. •
' 4 - sin 3a cos a '
whence

5m2

cos 2a. =

7
18 '

S ABC

=

m' tan 2a

=

Vn
7

224. The points C, M, D, and L lie on a circle,
consequently, LCML = LCDL = 30°. In similar fashion L C M K = 30°; thus, L LM K = 60°
and ~LMK is regular, I KM I = 21V5. By the
law of cosines, we find: cos L LC K = -3/5. Since
L DCB = LLCK - i20°, we have: I DB I =
2 - V3

vs

225. Let A be the point of intersection of the
straight lines Be and KM. The quadrilateral


fS4

ONBC is an inscribed one (LOCB = LONB =
90°), consequently, LOBC = LONC = a/2. Similarly, CMAO is also an inscribed quadrilateral and
LCAO = LCMO = a/2, that is, OAB is an isosceles triangle. Thus, I CB I = I A C I = I CO I X

V

+

a
a
a
cot -- =
R2
bl - 2Rb cos - cot 2
2
2·
226. The points E, M, B, and Q lie ona circle
of diameter BE, and the points E, P, D, and N
H

on a circle of diameter ED (Fig. 3). Thus, LEMQ =
LEBQ = t80° - LEDC
LEDN = LEPN,
analogously, LEQM = LENP, that is, the triangle EMQ is similar to the triangle EPN with
the ratio of similitude of
(For completeness of
solution, it is necessary to consider other cases of
the arrangement of the points.)
Answer: d Vii.
227. Extending the non-parallel sides of the
trapezoid to their intersection, we get three similar
triangles, the ratio of similitude of the middle to
larger triangle and of the smaller to middle one
being the same. Let us denote this ratio by A, the
larger base by x, the radius of the larger circle by
R. 'I'hen the line segments parallel to the larger
base are, respectively. equal to ~ and 11z , the

=

vii:

tSS


larger lateral side of the lower trapezoid to 2R.!!:..,
c

+

the second radius to ).R. Hence, R
).R = ~.
By the property of an circumscribed quadrilateral,

+M=

2R

+ 2R~.
c

=

2R c+d ,

And finally, dropping
.
from the end point of the smaller base of the entire
trapezoid a perpendicular on the larger base, we
get a right triangle with legs c, z - AZz, and hypotenuse d. Thus, we have the system
Z

I

x(1+)')

z(1 -11 ) =

Y dl -

: I,

R (t+1)=c/2,

whence ). =

d- V dl-c'
----c

Anaoer: the bases are equal to

and

d

d-Ydi="Ci
c

+ Ydi=Ci
e

228. Let us draw perpendiculars from the centres
of the circles to one of the sides and draw through
the centre of the smaller circle a straight line
parallel to this side. In doing 50, we obtain a right
triangle with hypotenuse R
r, one of the legs
R - r and an acute angle (1 at this leg equal to the
acute angle at the base of the trapezoid. Thus
R - r
cos l'£ = R
r' The ~arger base is equal to

+

+

2R cot

~

2R

2

equal to 2r tan

1/R.

"r

The smaller base is

;=2rY~,

t88


229. Let us take on the side AB a point K such
that I B K I = I BD I, and on the extension of
AC a point E such that I CE I = I CD I. Let us
show that the triangle ADK is similar to the triangle ADE. If A, B, and C are the sizes of the
interior angles of the ~ABC, then, LDKA =
t80° - LDKB = t80° - (900 - L B/2) = 90°
L 8/2, LADE = 1800 - L CED - LA/2 =

+

1800

~ (LA + LC) = 90 + LBI2. Thus,
= LADE. In addition, by hypothesis,
0

-

LA KD
LDAE

=

LDAK.

Auw,,: Ya'b.
230. Using the notation of the preceding problem, we have:
I AD II = (I AC I I CD I) (I AB I - I BD I) =

+

1+

IACI·IABI-ICDI·IBD
(I AB 1·1 CD I - lAC 1·1 BD I).

But the term in the parentheses is equal to zero

.
IABI
IBDI
.
smce I AC I = , CD f (see Problem 9 In Sec. t).

23t. Let us extend BN and eN to intersect the
second circle for the second time at points K and
L, respectively; I MN I = INK I since LANB =
900 and M K is a chord of the circle centred at A.
Since the corresponding arcs are equal, we have
LLNK = LBNC = LBND. Thus,
I LN 1=

I ND 1= b,
I MN 1·1 NK 1= I MN II
I MN I = ya;;:-

= ab,

232. Note that PQ is perpendicular to CB _ Let
T be the point of intersection of M Nand PQ,
and Land K the feet of the perpendiculars dropped
from C and B on the straight line M N (L and K
lie on the circles constructed on CN and BM as
diameters). Using the properties of intersecting
chords in circles, we get: I PT I-I TQ I = I NT I X
I LT I,
1PT 1·1 TQ I = I MT 1·1 TK I.
But.
I LT I = I CD I, I TK I = I DB I (since CLKS is
a rectangle and PQ is perpendicular to CB). Thus,

i87


IMT I I CD I
I NT 1·1 CD I = I MT 1·1 DB I, I NT 1= IDB I'

that is, the straight line PQ divides CB and M N
in the same ratio, hence, PO passes through the
point A, and D is the foot of the altitude.
Answer: I BD I ; I DC I = 1 :
233. Let LBOC = 2a, LBOL = 2~. Then
I A C I = 2R cos a, I CL I = 2R sin (a. P),
I CM I = I CL I cos (90 0 - P) = 2Rsin(a+ P)X
sin p, I AM I = I AC I - I CM I = 2R (cos a.sin (a. + P) sin P) = 2R cos P cos (a.
P), and,
finally, I AN I = a = I AM I cb's a = 2R cos a. X
cos p cos (a
P). On the other hand, if K, P,
and Q are the midpoints of AO, CO, and CL, re-

vS:

+

+

+

spectively, then I KP I = ~ I AC I = R cos C&.
Further I PQ I = 0R/2, L KPQ = L0 KPO
LOPQ = c£ + 180 - LCOL = 180 - a - 2p,

+

RI

and, by the law of cosines, I KQ 12 = T
R' cos' a
R' cos a. cos (a
2~) = R2/4 +
R"
2R' cos c£ cos p cos (a
P> = "4 + Ra .

+

+

+

+

.. J" R'
Answer: V T
n« ·
234. It follows from the similarity of the triangles M AB and M BC that
'MAl _ IMAI . IMBI _ IBAII _ ,
IMel - IMBI
IMCI - IBCls - k •
235. From Problem 234 in Sec. I, it follows
IAMls
IACI
IANI'
IADI
that IMBII ~ IBCI'
INBI' . IBDI - If K
is the point of intersection of M Nand AB, then
IAKI
SAlrlN
IAMI-IANI sin L MAN
IKBI = SBMN = IMBI·INBI sin L MBN .. /' IACI IADI
l/ r
a,p
V IBel· IBDI = Y (a·-t)C8-1)

+

188


236. Let K, L, M, and N be the points of tangency of the sides AB, BC, CD, and DA with the
circle, respectively. Let P denote the point of intersection of AC and KM. If L.AKM = q>, then
o

L.KMC = 180 - cp. Thus,

-} IAKI·IKMI sin q>

IAPI
IPCI =

IAKI

SAKM

SXMC -

a

..!.IKMI.MCI sin (180o - q» = IMel =T'
2

But in the same ratio A C is divided by the straight
line NL. Hence, the lines AC, KM, and NL meet
in the same point. Considering the diagonal BD
and reasoning in the same way, we prove that BD
also passes through the point P. The sought-for
ratio is equal to (JIb.
237. Let P and Q be the intersection points of
BK and AC, and AB and DC, respectively. The
straight line QP intersects AD at a point M t and
Be at a point N. Using the similarity of the cor-

..

responding triangles we get:

IAMI
IMDI

=

IBNI
INCI

=

IMKI _ IAKI-IAM'
If IAMI = IADI
IAMI IAMI
z,
then IAMI
IAMI
z
z
IMDI - IAnI-IAM' =.: t-z' 1-x =
A-X
A
- z - ' whence %= A+1
Anaoer:

t
i
If A=
then IAMI = n+1 IADI, Thus,
taking first K to be coincident with D (A = 1), we
get the midpoint of AB as M 1 ; taking K to be

n'


t89

coincident with M l..' we find that M I is 1/3 distant
from AD, and 80 forth.
238. Let I KM I = I KN I = e, I AD I = y,
and I DB I = I. Then I CD I = ,fYi, y
z = c.
The radius of the circle inscribed in the triangle
t
t ..r A KB is equal to 2 I CD I = 2' f 1/1. Express
the area of the triangle AKB by Hero's formulas
and S = pre We get the equation

+

y (z + , + I) z1l1 =
+

(z

+ , + I) ~

V;l.

Knowing that y
I = C, we find z = c13.
239. Through the point A I' draw a straight
line parallel 'to A c. Let R be the point of intersection of this line and A B. Bearing in mind that
'ARI
IBIA.I
1
IACtl
IRel1
IA,CII =T' ICtBt =k, we find:
I · 'I ar f as hilon, d rawIDg
"
n Simi
through Cia straight line parallel to AC to inIe
IARI -- (k+1)1
IABt

•

tersect BC at a point S, we obtain that

~~II =

(k; 1)1 • Therefore the points R, AI. CI. and
S lie on a straight line parallel to AC. Thus, the
sides of the tri~les ABC and AIB.C, are correspondiDgly parallel. Now it is easily obtained
that I AsCtl = IR81 - IRAsl - lelSI == t JiC I X

( 1.
18

(k~1)1

equal to

). therefore the ratio of similitude
kl-k+t
(k+1}1

240. Let us use the following formula for the
area of a triangle: 8 = 2RI sin .If sin B sin C, where
At B, and C are its. angles. Then the area
of the triang~e· Al~lCl' where A lt B t , and

too


Clare the intersection points of the angle
bisectors of the triangle ABC with the circumscribed circle, will be equal to 8 1 = 2RI X
· A+B . B+C . C+A 2R
C
sm --2- sin --2- sin
cos T X

-r= ·
A

8
cos -A cos -B an d 2
2 ' 81

sin

~

(cot
.
sin

• On the other hand, IBe I

~ +eot ~

B.
81n

2

= 8 . AsinB
sin - . 2
2

C

'2 ·

=

)=2RsinA, and
Th

S

US,

X

2R sin A, r X

r~4Rsin

: X

2r

8'; = If

241. Let 0 be the centre of similarity of the
inscribed and circumscribed triangles, M 1 and M
two similar vertices (M 1 lies on the side A B), an
let the line segment OA intersect the inscribed
triangle at a point K. Then SOM,K= 1S1 , SOM.A =
SOMtA
I OM I I , /"8';
1S t ,
S
. 10M I
V S ' whence

d

OM.A

I

I

K

SOM A =A VS1S., where A So;'t • ConsiderI
1
ing six such triangles and adding together their
areas, we get: S~BC = V SI S I 242. Let 0 denote the centre of the circumscribed circle, H the intersection point of the altitudes of the triangle ABC. Since the straight line
ON is perpendicular to the bisector of the angle A,
it intersects the sides AB and AC at points K and
M such that I AK I = I AM I. Thus, L AOB =
2 LC (we assume the angle C to be acute); LOAK=
900 - LC = LHAM. Hence, ~OAK = 6HAM,
and I OA I = I HA I = R (R the radius of the
circumscribed circle). If D is the foot of the perpendicular dropped from 0 on Be, then I OD I =
I AH 1/2=R/2. Consequently, cosA =C08 LDOC==
1/2, LA = 60°.

191

Answers, lIints, Solutions

243. Prove that the triangle will be acute, right,
or obtuse according as the distance between the
centre of the circumscribed circle and the point
of intersection of the altitudes is less than, equal
to, or greater than half the greatest side,
AlUwer: 90°, 60°, and 30°_
244. The condition S ~BDM = S ABCK means
that I BD I-I BM I = I BK 1·1 BC I, that is,
(1)

(lBAI+IACDIBMI=IBK(-lBCI-

Through M, draw a straight line parallel to A C;
let L be the point of intersection of this line and
BA. Prove that I LM I = I XL I; hence it follows

that the desired LBKM= ~ LBAC= ; . Since
the triangle BLM is similar to the triangle BA c,
IBMI

= /BCI-IACI,

we have ILMI

"IBMI
=l'BCf X

IBLI

IABI- Now, we find IBK( from (1) and compute:

=

IKLI

=

IBKI -IBLI

IBMI

+

IBAI
IACI. IBM( IBCI

(BMI

IBCI • lAB I = (Bel·ACI, whence I LM I =

(XLI.

245. Let IADI =4, IBCI =b. Drop from 0 a
perpendicular OK on AB. We now find: lBKI =

.. rt:
f

b

ab b+a'

I

BE I

Yab
..rt:
b
~-f ab b+a
../ -

=

/-

b

J ab a-b

..
=,rt:
ab

a-b
2(a+b)

2ab

t

I

=

IEKI

1

=

I

=

MK

tBEI

+ IBKt =

a~b

• It is easy to check that IOKllI= IEKI·IMKI.

An.a.uer: 90°.

f

ab- (a-b) (a-t-b) ,

10K


t92

246. Note that the points At M, s, and 0 lie
on the same circle (see Fig. 4). Consequently,
LNMO = LOAN = 000 - LAON. Hence, with
OA rotated about 0 through an angle
the straight
line N M rotates through the same angle cp (in the
opposite direction), and when A displaces along

,J,

0 ....- - - - - -.....- -

Fig. 4
OA, the line N M displaces parallel to itself. Hence
it follows that the desired angle is equal to CI.
247. If 0 1 is the centre of the smaller circle and

"BOA='P, then "BAO=900-;, LCOIA=
90"

+ 'P,

LBAO -

LCAOI

=

45° -

; • Thus, "BAC =

LCADl = 45°.

248. Construct a regular triangle ABK on AB
inaide the square. Then L KAB = 600, L KCD =
15°, ·that is, K coincides with M.
A nBU1~r: 30°.
249. Let M 1 be symmetric to M with respect to
Be and CB is the bisector of the angle MCM 1 •

Since "MICA = 60" and I AC I = ~ I CM I I we
have that LM1A C = 90°, hence AB is the bisector

193


of the angle MIA C. In addition, CB is the bisector
of the angle M 1C M, that is, B is equidistant from
the straight lines M IC and MIA and lies on the
bisector of the angle adjacent to the angle AM IC.
Thus, LBMC == LBMIC == 75°.
Answer: 75°.

250. If L BA C == 2a., then we readily find that
LKMC = LMKC == 30°
a., that is, I Me 1=
I KG I. Let us extend M K to intersect the circle
at a point N; 6.KMG is similar to 6.KAN, hence,
I AN I == I KN I = R, i.e., to the radius of the
circle (since L.AM N = 30°). The points A, K,
and 0 lie on a circle centred at N, LANO == 60°,
consequently, LA KO == 30° or 150° depending
on whether the angle A MC is obtuse or acute.
Answer: 30° or 150°.
251. (a) Draw the bisector of the angle A and
extend BM to intersect the bisector at a point N
(Fig. 5). Since I BN I = I sc I, LBNG == 120°,

+

A

c

lJ

Fig. 5
hence each of the angles RNA and CNA is also
equal to 120°, LNCA = LNCM == 20°, that is,
~NMC = ~NCA, I Me I == I AC I. Consequently, the triangle AMC is isosceles, and LAMC ==
70°.
(b) The points M, P, A, and C lie on the same
circle (the point M from Item (a»; LPAC ==
LPMC == 40°.
13-01557

t94


252. Describe a circle about the triangle MCB
(Fig. 6) and extend BN to intersect this circle at
a point M.; I CM. I == I CM I since the sum of
the angles subtended by them (80 and t 00°) is
equal to 180°; LM1CM = LM1BM = 20°, that
0

A

c

C
Fig. 6

Fig. 7

is, NC is the bisector of the angle M1CM and
6.M 1CN == ~NCM,
LNMC == LNM1C =
LCMB = 25?
253. On Be, let us take a point K (Fig. 7)
such that LKAC = 60 0 , M K II AC. Let L be the
intersection point of AK and MC; ALe is a regular
triangle, AN C is an isosceles triangle (the reader
i~ invited to determine the angles). Hence, LNC


195

is also an isosceles triangle, L LC N = 20°. Let us
now find the angles N LM and M K N -each of
them is equal to 100°. Since M K L is a regular
triangle, each of the angles KLN and N KL is equal
to 40°, Le., I KN I = I LN I and ~MKN =
~MLN, LNML = L.KMN = 30°.
254. Let us take a point K (Fig. 8) such that
LKBC == LKCB == 30° and denote by L the
intersection point of the straight lines MC and BK.

A

C

Fig. 8
Since b.BNC is isosceles (LNBC = LNCB =
50°), L KNC = 40°. L is the intersection point
of the angle bisectors of the triangle N KC (LK
and LC are angle bisectors). Consequently, NL
is also the bisector of the angle K N C and L LNB =
60°; BN, in turn, is the bisector of the angle MBL;
in addition, BN is perpendicular to M L; hence,
BN bisects ML, and LMNB = LBNL == 60°
and LNMC = 30°.
255. Let 0 be the centre of the inscribed circle;
the points C, 0, K, and M lie on the same circle
(L COK = L. A/2
L C/2.= 90° - L B/2 =

+

13*

196

LKMB = 180° - LKMC; if the point K lies
on the extension of NM, then LeOK = LCMK).
Thus, LOKC = LOMC = 90°.
256. If P lies on the arc AR, Q on the arc AC,
then, denoting the angle PAB by (J), and the angle
QA C by '1', we get two relationships:
C - cp),
sin2 (C - cp) == sin q> sin (B
{ sin 2 (B - 11') = sin", sin (B
C - 'l').
Writing out toe difference of these equalities and
transforming it, we get: sin (B
C - q> - '1') X
sin (B - C) + (cp - 'l')]=sin (B
C - cp-",)X
sin (cp - '1'), whence (since 0 < n) B - C + cp - "I' = :t - (cp - '1') and we
get the answer.

+
+
+
+

+

n-a

Answer: - 2 257. Let us prove that the triangle eMN is
similar to the triangle CAB (Fi'g. 9). We have:

CF------+--+-~--_14

LMCN = LeBA. Since the quadrilateral CBDM
·
.
ibed
h
ICMI sin L CBM
is an insert
one, we ave ICBI
sin L CMB=

sin L CDM
sin L DBA
IADI
I CNI
sin L CDB = sin L ADB =~=lAIiI·

Hence,

197

L CM N = L BCA, that is, the desired

ith
a
a
·
angIe IS equa 1 to eit er 2" or n- 2
258. Let LABC = 120°, and BD, AE, and CM
the angle bisectors of the triangle A Be. We are
going to show that DE is the bisector of the angle
BDC, and DM the bisector of the angle BDA.
Indeed, BE is the bisector of the angle adjacent
to the angle ABD, that is, for the triangle ABD,
E is the intersection point of the bisectors of the
angle BAD and the angle adjacent to the angle
ABD; hence, the point E is equidistant from the
strairht lines AB, BD, AD; thus, DE is the bisector 0 the angle BDC. Exactly in the same way,
DM is the bisector of the angle BDA.
259. Denote: LABD = a, LBDC = <po By
hypothesis, LDAC = 120° - a, LBAC = 30°+
a, LADB == 30 0 - a, LDBC = 60° a. By
the law of sines for the triangles ABC, BCD, ACD,
we et IBCI _ sin(300+a} _
1
g
IACI - sin (60o+2a.) - 2cos(30o+a) ,
IDCI
sin (60° +a)
I ACI
sin (30'J - a +cp)
IBCI ==
sin cp
'IDCI == sin (1211°-a) •
Multiplying these equalities, we have: sin (30
a+cp) ==2cos (30°
a)sin<p~2cos(60°
(X) X
sin (300 - <p) == 0; thus L BDC =~ cp == 30° .
260. In Problem t 7 of Sec. 1 we derived the
formula for the bisector of an interior angle of a
triangle ABC. In the same way it is possible to
prove that the bisector of the exterior angle A is

+

0

+

+

2bc sin

computed by the 'formula LA=
C,

I BC I = a,

sin : : sin :

I CA I

= b).

-

~

Ib _ c I (I AB I =
We

then

= VI" +(t-COSA) =

find

198


.. / 1. ( _ bt + Ct V

2

1

a2 )

2bc

_

--

..

/(a+b-c) (a+c-b)
4be
·

V

.
" A d SID C ·
"
F Indimg In th e same way I e SIn T an · 2 In
terms of the sides of the triangle, and equating
Vc(a+b-e)
Va (b+c-a)
lA to lc. we get
I b- cl
- Ib-al
·
By hypothesis, b = 2, c = 1. Hence, a must satisfy
../ Va(3-a)
the equation" a+1=
la-21 =9 (a - 1) X
(a2 - a - 4)= 0. But a =1= 1, consequently, IBGI=a=

1+Vi7
2
261. If 0 and 0 1 are the centres of the circles
circumscribed about the triangles ABC and ADB,
respectively, then the triangle A 001 is similar
to the triangle ACD.
Answer:

«n.

262. If K is the midpoint of the arc A B, 0 the
centre of the circle, I AB I = 2R = c, then
I

cu 12 =

I CD 12

+ I DM)2==1 CD 1 + I DK
2

2

1

2
I AD I· I DB I + R2 + I DO 1 == (R + I DO I)

+

+

=
X

(R - 1 DO I)
R2
I DO pi = 2R2 ==
/ 2.
Answer: c V2i2.
263. Let KM be a line segment parallel to BC,
and Nand L the points at which the inscribed
circle touches the sides A C and BC. As is known
(see Problem 18 iT Sec. f), 1 AN I = I AL I ==
p - a, where p is the semiperimeter of the triangle
ABC. On the other hand, I AN , = , AL I is the
semiperimeter of the triangle A KM which is
similar to the triangle ABC. Consequently,
p-a
b
a2
----- p--p - a' - a-b ·
Answer:

2a1
--J..- •

a-v

c2


199

264. If a, b, c are the sides of the given triangle,
then the perimeters of the cu t-06 triangles are
2 (p - a), 2 (p - b), 2 (p - c), where p is the
semiperimeter of the given triangle. Consequently,
if R is the radius of the circumscribed circle, then
p-a
p-b
P-C)
--+--+-p
p

R 1+R,.+R3 = ( p
Answer: R 1
R2

+

+ R a•

R~R.

IACI
265. If L A=a, then IAMI -=-.-, IAN/=
sin a.

I~BI ,
sin a
~

that is, IAMI:IANI

= IACI:IABI;

thus,

AMN is similar to ~ ABC with the ratio of

similitude - .f_, therefore 1MN I == I ~C I

== 2R.

sin ex
srn c
266. Let 0 1 and 0 1 be the centres of the inter-

secting circles. We denote their radii by x and y,
respectively, I OA I = a. Since, by hypothesis,
the triangles AOO. and ADO! are equivalent, expressing their areas by Hero's formula and bearing
in mind that lOlA I = x, I 001 I = R - e,
lOlA I = y, I 00 2 I = R - y. after transformations we get: (R - 2x)2 = (R - 2y)2, whence
(since x =1= y) we obtain: x
y == R.
Answer: R.
267. Let A B and CD be the given chords and M
the point of their intersection.
(a) The sum of the arcs A C and BD is equal to
fBO° (semicircle); consequently, lAC 12 + 1 BDI2=
4R2,
thus,
I AM 12
I MC 12
1 MB 12
1 MD 12 = lAC 12
I BD 12 = 4R2.
Answer: 4R2.
(b) lAB 12
I CD 12 = (I AM I
1MB 1)2 +
(I CM I I· MD 1)2 = 4R2 + 2 I AM I I MB 1+
2
2 I CMI· I M D 1 == 4R2 + 2 (R2 - a ) = 6R2 - 2a2.
Answer: 6R2 - 2a2 •
268. If M is the second point of intersection of
Be and the smaller circle, then I BM I = I PC 1

+

+

+

+

+

+

+

+

200


I BP I = I MP I + I BM I,
2
2
2
2
I PA 1 + I PB 1 + I PC 1 = I PA 1 + (I PB I -

(M betweenB and P),

+

+

+

2
2
I PC 1)2
2 I PB 1·1 PC I == I PA 1
IMP 1
2 I PB I I PC I = 4r2 + 2 (R2 - r2 ) == 2 (R2 r2 ) .
269. Let us denote the lengths of the segments
of the chords as in Fig. to and the diameter by 2r.

Fig.

+

to

Taking advantage of the fact that the angles based
on the diameter are right ones, and zy = UV, we
get z (z + y)
u (u + v) == (u
v)2
x 2 -v2=
(u
V)2
m 2 = 4r2 •
270. If a, p, 'V, 6 are the arcs corresponding to
the sides a, b, c, and d, then the equality to be
proved corresponds to the trigonometric equality

+

+

+

+

+

. a cos- cos -a SID - -SID - cosv+
.y
.~
6+
2
2
2
22
2

SID -

. a+v
. P+6
or SID --2- = SID --2271. Let A BCD be an inscribed quadrilateral.
A B and CD intersect at a point P, A and D lie on
the line segments BP and CP, respectively. BC
and AD intersect at a point Q, while C and D lie
on the line segments BQ and AQ. Let us circumscribe a circle about the triangle AD P and denote
by M the intersection point of this circle and the
straight line PQ. (Prove that M lies on the line
segment PQ.) We have: LDMQ = LDAP =
LBCD. Consequently, CDMQ is an inscribed
quadrilateral. Since, by hypothesis, the tangents
drawn from P and Q to the original circle are equal
to a and b, respectively, we have I QM 1·1 QP I =
cos

~ . 6
"2 SID 2"


201

I QD t·1 QA 1= b2 ,
I PM 1·1 PQ' = I PD I X
I PC I = a2 • Adding together these equalities,

+

bi •
we get I PQ 12 = a2
2
Answer: V a
v.
272. The line segment QP is equal to
Y(b 2 - R2)
(c2 - R2) = Vb 2
c2 - 2R2 (see
the preceding problem). Let ABeD be the given
quadrilateral, Q the intersection point of A Band
CD (A lies on the line segment BQ). To find the
length of PQ, we circumscribe a circle about the
triangle QCA and denote the point of intersection
of QP with this circle by N. Since L.ANP =
LACQ = LABP, the points A, B, N, and P
also lie on a circle. We have I QP 1·1 QN I =
I QA 1·1 QB I = b2 - R2, I PN I·IPOI = lCP tx
I PA I = R2 - a2 • Subtracting the second equality from the first one, we get I QP 12 = h2
a2 2R2. Analogously, I PM 12 = c2
a 2 - 2R2.
Answer: I QM I
Vb2 c2 - 2R2, IQPI =
2 +a - 2R' , IPMI=Vc2+a2-2~2.
2
Yb
273. The radius of the inscribed circle is contained between the values of the radii for the two
limiting cases. I t cannot be less than the radius
of the circle inscribed in the triangle with sides
a
b, b
c, c
a which is equal to
where
S is the area and p the semiperimeter 0 the tri-

+

+

+

+

+

angle;

+

thus

V a+a:c+ e'

+

+

81'

+

r

>

S
v:

V (a

+ b + c) abc
a+b+c

On the other hand,

r

=

is less

than the radius of the circle shown in Fig. 11 (on
this figure, the opposite tangents are parallel, and
the point C tends to infinity). Since for the angles
a, ~, and y marked in the figure the following
equality is fulfilled: a
Ii y = 1t/2, tan a. =
clp, tan p = alp, tan y = hlp, where p is the
radius of the shown circle, tan (a
~) = cot y,

+ +

+

202


or (e+p)p
pi- ae
Thus,

Pb' whence p

==

Vab

+ be + ca.

,;r - abc ~/
V a+b+c < r < " ab+bc+ca.

A a
0-------

IJ

D
c

Fig. 11
274. Let M be the point of intersection of the
straight line CB and the lines of centres of the
given circles. Let us denote: I A M I = z; LA CB =
q>; I AB pi = 2rz, I AC IS = 2Hz, sin q> = I AZC "
If P is the radius of the circle circumscribed about
AABC ,en P == 2·
th
IABI = lAB 2 AC I = ..rr .
1·1
u.
yR
SID cp
x
A nsuier: V Rr .
275. Let
O 2 be the centres of the circles and
A the point 0 their intersection most remote from
BC, LOlAO, = <p. Let us show that LBAC =
I

01'

q>/2. (For the other point the angle is 180 0
0

-

~

.)

Indeed, LBAC = 180
LABC - LBCA =
t80° - (900 - LABOt ) - (900 - LACO,) =
LABDt
LACO s = L.BAOl + L.eAO, = q> '-BAC). Let 1°1 o, I = a.~Drawing O.M II Be
-

+

(M on

V as -

203

° l B ),
we get I BC 1 = 10 2M I =
(R-r)2. From the triangle 0lA02 we find

+

R?
r 2 - a2
that cos cp =
2Rr
; thus, the radius of
the circle circumscribed about the triangle ABC
is equal to

Va -

IBel

2

--~

2sin -!.
2

}/2

(R - r)2

J/1- R2+ r
2Rr

2-

al

VRr.
A nsuier; V Rr (for both triangles).
276. DO and CO are the bisectors of the angles
ADC and DeB. Let a, P, and y denote the corre-

0

A

6

Fig. 12

+

spending angles (Fig. 12). But ex
2~ + 21' +a =
2n; hence, a
p + y = n; hence it follows that
LDOA = ," LeOB = p, and the triangle AOD
is similar to the triangle COB; consequently,

+

I AD 1·1 CB I == lAO 1·1 OB I = lAB 12 /4 .
Answer: a2/4b.

277. It follows from the conditions of the
problem that the bisectors of the angles C and D
intersect on the side A B. Let us denote this point
of intersection by o. Circumscribe a circle about
the triangle DOC. Let K be a second point of
intersection of this circle with AB. We

have: LDKA

=

LDCO

= ~

LDCB

=

~

X

204
(1800

LDAK)

~

(LDKA

+

LADK).

Hence, LDKA = L.ADK and I AD I = I AK I.
Similarly, I BC I = I BK I; consequently, 1AD 1+
1 CB I = I AB I.
Answer: a-b.
278. On the ray Me, we take a point N such
that I AN I = I AB I = I AD I. Since
sin L MNA
I AC I
lAC 1
sin'LADC
sin L MCA = I AN , = I AD I = sin" L ACD
and L MCA = LACD, we have: sin L. MNA
sin LADG = sin LABM, that is, the angles
ABM and MNA are either congruent or their
sum totals to 1800 • But M is inside the triangle
ABN, hence, L.ABM == LMNA. Now, we can
prove that ~ABM = ~AMN; LNAC ==
LMNA - LNCA = LADC - LACD = cp.
a+q>
Answer: -2-.
279. Let K and L denote the points of tangency
of the first and second circles with one of the sides
of the angle, and M and N the other points of
intersection of the straight line AB with the first
and second circles, respectively. Let 0 denote the
centre of the second circle. Since A is the centre
'1 I AL I
oI si ilarit yo I the grven circ es, IAKI = IAMI
simi art
ie zi
I AB 1==

: ~:II = 1, whence I AK ',1 AL 1= 11 AL

12 =

2
I AB 1·1 AN I = I AB 1 • On the other hand,
from the similarity of the triangles A KC and ALO
we have: I AK 1·1 AL I = lAC 1·1 AD I. Con2
sequently, lAC 1·1 AD I = lAB 1 ; hence, the
triangles ABC and AOB are similar.

it,

Answer:

a

"'2

or n -

a.

2·

280. Let LBAF = cp, LDBA == a, LDAB ==
2a (by hypothesis, it follows that the points A, E',
and F lie on the same side of BD, and L.BDA

<

205


90°, that is, a. > 30°). By the law of sines, for the

.

IDEI

triangles DEA, DAB, and BAF we have: IADI
sin (120"-2a)
sin (300 + a )

=.::

2 cos (30o + a )" IADI= ~inCL =
, IABI Sin 3ct,

IABI
IBF I

1
4 cos (300 + a) cos (300 _ . a) ,

=

cos (a-«p)
sin <p.

Multiplying the equalities, we find: cos ~a-q» _
sin q>
2 cos (a-30°), whence L BAF=q>=30°.
281. Consider two cases.
(1) The line segment BI( intersects AC. From

the condition that L.BKC = 3 L.A ;- L.C it fol-

= LB + LC,
+ (LB + LC) +

lows that LC = 90° (LBCK
LCBK = L
3 L.A ; : LC
2B,

~B

Consequently, the point 0 is
found on A B, and the sum of the distances from 0
to AC and AB is equal to ~ I BC I; thus, I BC 1=
= 1800, etc.),

+

+

4 > 2
Y3 = I A C I I A B I > I A B I,
that is, a leg is greater than the hypotenuse which
is impossible. Thus, we have arrived at a contradiction.
(2) The line segment B K does not intersect A C.

In this case, L CBK = 180°,A, LBKC =
(180

L B,
2

3 L A-L C
.
2
(by hypotheslsuhence,

+

A+

L
3L
2B)
.whence L. A = 30°.
Again, two cases are possible.
0

-

L

L BCK =

A; L

C

,1800,

206


(2a) The centre of the circumscribed circle 0
is inside the triangle ABC. Let the perpendicular
dropped from 0 on A B intersect A B at N, and A C
at K, and let the perpendicular drawn to A C intersect AC at M and AB at L. Let us denote: IOMI==
x;
I ON I = y; x
y = 2 (by hypothesis),

+

lOKI
2 INK

= 2x1V3, IMKI = x1V3, IAKI =
I = 2y + 4xl V3, I A M I = I A K I

I M K I = 2y

+

2x

~

y

(I AB I

+ x1V3. Similarly, we find:

Va:

By hypothesis, I AN I

+' I

AC I)

= ~

(2

+

IANI=

+ IAMI=

V3). On the

+

other hand, r AN I + I AM I
(2
}-13) X
(x
y) = 2 (2
Va), which is a contradiction.
(2b) The point 0 is outside the triangle ABC.
We can show that L B is obtuse. Otherwise, if
3 LA - LC
.
L C > 90°, then
2
< 0, thus, 0 IS

+

+

found inside the line segment A C not containing B;
however t this does not affect the answer. Using
the notation of the preceding item, we have:
I AM I = 2y - z V3, I AN I = .y V3 - 2%
From the system y
x = 2, I A M I
I A N I :::I

+ V3) y -

(2

_

3

+
(2 + Va)
5

+

%

=

V 3 we
2 +2
5

find:

3 y"3

Y = - , I AM 1= - - - - , the
4
2
4
2
radius of the circle is V I AM 1 + I MO 12 =
% -

t/2

-,

4

Y34 -15 V3.

282. If C1 is a point symmetric to C with respect
to AB, and B 1 is symmetric to B with respect to
AC, then (as usually, a, b, c are the sides of baABC,
S its area) I CtB t 12 = b2 + c2 - 2bc cos 3A =
a2
2bc (cos A - cos 3A) = a2
Sbe sin 2 A X
82
cos A = a2
t6 (b2
c2 - a2 ) b2c2t Thus, weget

+

+

+

+

Answers, Hints. Solutions

207

the system of equations:
a2b2c2
168 2 (b2 + c2 - a2) = 8b 2c2 ,
a2 b2c 2
168 2 (a 2
b2 - c2 ) = 8a 2b2 ,
{ 2 2c 2
ab
168 2 (e2
a2 - bl ) = 14e2a2 •
Subtracting the second equation from the first one
and bearing in mind that a =1= c, we find: 48 2 =
bl • Replacing 8 1 by bl/4, we get:
a2e2 + 4 (b2 e2 a2 ) = 0,
alb2e2
4b 2c2
4b 2aS - 4bt - 14a2e2 = 0,

+
+
+

{

+
+

+

+

b2 = 4S2 •

+

Denoting a2e 2 = x, a2
c2 = y, we have:
4y - x == 4b2 ,
{ X (b2 - 14)
4b2 y = 4bt .
Multiplying the first equation of the latter system
by b2 and subtracting the result, from the second
equation, we find: x (2b2 - 14) = 0, whence b '=

+

Y7".

Answer:

V7,

-V

1,

../- .f /- or 1/' 21- 2V217
f 7,
8

t

21+r 217

I b2

-

e2

I

283. Prove that tan a =
28
t where S
is the area of the triangle (prove this for the other
angles in a similar way).
A nsurer: arctan , tan a ± tan PI.
284. Let us find the cotangent of the angle
between the median and the side of the triangle
ABC. If LAtAB = q> (AA 1 a median of the triangle ABC, a, b, e the sides of the triangle, m a ,
mb, me its medians, 8 the area), then cot <p ==
2c - a cos p _ 2c2 - at cos B _ 3c2
h2 - a2

+

he

-

28

-

48

•

Let M be the median point of the triangle ABC;
the straight lines perpendicular to the medians

208


emanating from the vertices A and B intersect at CI ;
L MC1B= L MAB=q>(MAC1B is an inscribed quad-

.

rilateral). Consequently,
(2a 2

5MBC

+2c - 728 +b b
2

2

2)(3c2

I

=

1(2
'2 3' mb )2

X

a2 )

co t q>
· The area of the
required triangle is the sum of the areas of the six
triangles, each area being found in a sim ilar way.
.
(a2
b2
C2)2
27 (R2 - cl2)2
Finally, we get
128
=
48
(the equality a2 + b2 + c2 = 9 (R2 - tJ2) is left
to the reader).
27
Answer: '4 (R2 - d2)2.

+ +

285. 60°.
286. First note that I M N I is equal to the
common external tangent to the circles with centres
at 0 1 and O2 (Problem 142, Sec. 1). Consequently,
if the radii of these circles are x and y and x
y =
2R - a, then I MN I = Va 2 - (x - y)2. Let q>
denote the angle formed by AB with 0 10 2 , L the
point of intersection of AB and 0 10 2 , We have:

+

lOlL

I= x

2R a

za

a

R

2R - a I 2x

+y=

xa.

2R _ a' sin q>

I OL I

+a -

Ix
2R I

V R2

+

=

x

=

lOlL

lOlL

I-

R

2R _

_4

IZ

R
-

I

I
y

I,

- I OL
q> =
2R
(X-y)2 = - I MN I.
a
Answer: 2R (in both cases).
a
287. The angle A K B is equal to 900 (see Problem
255, Sec. 1). Let R be the point of intersection
of BK and ACt Q a point on BK such that NQ IIAC.
Using the usual notation, we have: IAR I = IAB,=
c, I MR I = c - (p - a) = p - b = I NB I,
I AB I
2R ,r 2
J' a a

2

2
1

sinl

209


I MK I _ I MR 1_ 1CB I
a
I KN I - I QN I - I Re I
b - c (b > c).
Since I MN I
2 (p - c) sin ;, I MK 1=
a

sin ~. Other line segments are considered in a

similar way. The desired triangle is similar to the
triangle ABC, the ratio of similitude being equal
to sin (a/2). Its area equals S· sin2• (a/2) •
288. Let I AM I = x, I eN 1= y, x
y = a,
where a is the side of the square. We denote by E
and F the points of intersection of MD and DN
with AC. The line segments I AE I, I EF I, I CF I
are readily computed in terms of a, e, y, where2
upon it is possible to check the equality I EF 1 =

+

I AE 12

+ 1 FC 1

2

-

I AE I-I FC

I.

289. Let P be the point of intersection of the
straight line DE with AB, K a point on AB such
that KD is parallel to A C, A KD is an isosceles
triangle (LKDA = LDAC = LDAK). Hence,
XD is a median in the right triangle, and I M N I =

1. I KD I =.!
2
4

I AP I

= .!
4

I AE 1 =

.!
4

a.

290. Let A I be another point of intersection of
the circles circumscribed about tha ~ABC and
~ABIC;. It follows from the hypothesis that
I BB t I == I cc, I, in addition, LABA I == LACA 1
and
LABIAl == LACtA t.
Consequently,
6A 1BB1 = ~AICCI· Hence, I AlB I = I AIC I·
Let LABC =~, LACB == "'I, LABA I ==
L.ACA 1 = cpo Since ~AIBC is isosceles, we have
LA1Be == LAtCR, Le., p
cp == V - <p, 

and if the radius of the circle circumscribed about the b,.ABC is R, then I AA I I =
2R sin y ;- P; but I AB I
I AC I
(y -

2R (sin y - sin
14-01557

p> =

4R sin y ;-

t

p cos P

y


210

a
2;

sin

consequently,

rAA

1

I

a

a

2 sin '2
291. Note that the points A, 0, M, B lie on the
same circle (L AMB is measured by one-half of the
sum of the arc A B and the arc symmetric to A B
with respect to DC, that is, LAMB = LA OB).
We layoff on AM a line segment M K equal to MB;
then the triangle A KB is similar to the triangle
OMB.
Answer:

I AB I == 2a.

292. Let I AB I == 2r, I Be I = 2R, 0 1 the
midpoint of AB, 02 the midpoint of BC, 0 3 the
midpoint of A C, 0 the centre of the fourth circle
whose radius is x, From the conditions of the
problem it follows that I 0 103 I = R, I 0 20 3 I =
r, I 010 I == r
x ; I 0 20 I == R
x, I OsO I =
R + r - x, Equating the expressions for the areas
of the triangles 0 10 0 3 and 0 100 2 obtained by
Hero's formula and as one-half of the product of
the corresponding base and altitude, we get two
equations:

+

.• /

y (R+r) r (R-x) x==

+

21

Rd.

{ Y(R+r+x) Rrx=+ (R+r) d,

Squaring each of them and subtracting one from
the other, we find: x = d/2.
Answer:

sn.

293. Let P be the foot of the perpendicular
dropped from N on the straight line M B, then
IMP I = R cos a; consequently, , M P I is equal
to the distance from the centre 0 to A B. But the
distance from the vertex of a triangle to the point
of intersection of its altitudes is twice the distance
from the centre of the circumscribed circle to the
opposite side (Problem 20, Sec. 1), Le., I .MP I =


211

f

I M K I· Hence, it follows that if M is located
on the major arc, that is, LAMB = a, then
I NK 1= R; and if L.AMB = 1800 - a (that is,
M is found on the minor arc of the circle), then
INK r~ = R2 (1
8 cos? a).
Answer: I NK I = R if M is on the major arc
and INK I = R V 1
8 cost a if M lies on the
minor arc of the circle.
294. Let ABC be the given tr-iangle, CD its
altitude, 0 1 and 0t the centres of the circles inscribed in the triangles A CD and BDG, respectively, K and L the points of intersection of the
straight lines DOl and DO'}, with AC and CB,
respectively. Since the triangle ADC is similar to
the triangle CDB, and KD and LD are the bisectors
of the right angles of these triangles, 0 1 and
divide, respectively, XD and LD in the same
ratio. Hence, KL is parallel to DID". But CKDL
is an inscribed quadrilateral (L KCL = L KDL =
90°). Consequently, L CKL = LCDL = n/4,
L CLK = L Cll K = 11,/4. Thus, the straight line
0/02 forms an angle of 11,/4 with each of the legs.
I M and N are the points of intersection of 0 102
with CB and AC, then the triangle CMO t is congruent to the triangle cno t (C0 2 is a common side,
LOtCn = LOtCM, LCD0 2 = LeMO t ) . Hence,

+

+

°'},

I

cu I =

I NC I

=

h.

Answer: the angles of the triangle are 1t/4, n/4,

n/2, and its area is h2/2.

295. For designation see Fig. 13. CKDL is a
rectangle. Since L LKA = 90° + a, LLBA =
90° - a, BLKA is an inscribed quadrilateral,
I LC I
h cos a,
1.
tan q> === ICAT =
h
== T sm 2 a.
(1)
sin a,

11 R is the radi us of the circle, then
'R ~~i

-

:-14*

IKLI
2 sin q>

h
2 sin q> •

(2)

212
Since L LOX

2 h

== 2q>,

. h 2a

tan <p
(1) and

SID

we have: IONI = Rcoscp

==

(we have used the equalities

(2», I OM I

=

I ON I sin (90° - 2a.) =

C
p

/(

N a

r;
Fig. 13

:~:;:

h

h cot 2a, and, finally, we get the ex-

pression+ I PQ I
... /

V

h
h

h"

= I QM I =

VRI -

10M II =

h2 cot l 2 a =

4 sin! <p

V+

(1 +cot l cp) - COl l 2a =

V ~ ( + Sin~
1

2a ) -

cotl 2a

=

h

rs

IPQI =h VS. If now the segments I PD I and
IDQI of the chord are denoted by x and y, then
x+y = h V5, ZII=h 2 , whence the desired line

segments are equal to

V5+1
2

213

V5-1
2

h,

h.

296. Let (Fig. 14) P and Q be the points of
tangency of the tangents drawn from E. Prove that

8

C ....---.......----+--~E

,
A

If

N

Fig. 14
2
I EP I = I EQ 1 = I BD I. Indeed, I EP 1 =
2
(I ED 1+ I DC I) (I ED I - I DC I) = I ED 1 -

I DC 12 = I Be 12

-

I DC 12 == I BD 12

(by

hy-

I ED I = I Be I). Denote I KN I == e,
I PN I = INA I=Y· I EQ 1==1 EP 1=1 BD 1=:.
Then I KE I = x + y - s. We have: SKEN =

pothesis,

1
"2 x (2R-z);
SKOE -

R (x -

on the other hand,.SKEN=SKON+
1
SEON = 2 R (x
oX
y - : - y - a) ==

+ +

1
s). Thus, 2":t (2R-z)==-R (x-z), :t=-2R.

Answer: 2R.

291. First, find lim IAOI
a .. O

lOCI ·

Denote: L

C=p.

214


We have:

1

IAOI
lOCI

SABD

=

SBDe

= ~

2" a

b.
sma

(1)

(p-a) (p- b) sin p

+

But by the law of cosines, a2 bl - 2ab cos a =
(p-a)2+ (p- b)2- 2 (p- a) (p- b) cos P=> cos p=
p(p-a-b)+abcosa
h
(p-a) (p-b)
, w ence

+

sin P = V 1 - cost P = V (1 - cos P) (1
cos ~)
Vab (1 -cos a,) (2p2- 2ap- 2bp+ab+ab cos a)
(p-a) (p-b)
(2)

If

a

~

0,

then

cos a.

~

1;

consequently,

sin a
VZ cos a -+- Vi as a -+ O. Taking
2
1-cosa
this into account, we obtain from (1) and (2):
V

1·

IAOI

al~ lOCI =

~.:no I

"11 / '

V

ab
S·
lAC
(p-a) (p_ b) • mce
I-+p,
pVab

AO

1= Vab+ V(P--a)

(p-b)

Section 2
t. Prove that if D is the projection of M on A B,
2
then I AD 1 - I DB 12 = I AM 12 - 1MB 12 •
2. Suppose that there is such a point (let us
denote it by N), then the straight line M N is perpendicular to all the three sides of the triangle.
3. If M is the point of intersection of the perpendiculars from Al and B 1 on Be and AC, then
(see Problem 1 in Sec. 2) I MB 12 - I MC II =
2
2
I AlB 1 , _ I Ale 1 ,
I Me pa - I MA 12 =
I BIC 12 - I B 1A 12 ; adding together these equal-


215

ities and taking into consideration the conditions
of the problem, we get: I MB 12 - 1 MA p~ ==
I CIB 12 - I CIA 12 , that is, M lies 011 the perpendicular drawn to AB through C .
4. It follows from the result of the preceding
problem that the condition of intersecting at one
point for the perpendiculars dropped from A I' B I'
and Cion the sides Be, CA, and A B is the same
as that for the perpendiculars from A, B, and C
on BIC., CIA t , and AlB., respectively.
5. We note that the perpendiculars dropped from
Alt Bit Cion BC, CA, AB, respectively, intersect at one point D and then use the result obtained
in the preceding problem.
6. The next problem proves a more general
fact. From the reasoning of that problem it will
follow that the centre of the circle lies on the
straight 1i ne A B .
7. We introduce the rectangular coordinate
system. If the coordinates of the points AI, A I' . .,
A n are, respectively, (Xtt Yt), (XI' Y2)'· ., (x n , Yn)
and those of the point M are (x, y), then the locus
y2)
be
cy
is given by the equation a (%2
d = 0, where a == k l + k 2
+ k n ; hence,
there follows our statement.
8. If B is the point of tangency and 0 the centre
of the given circle, then 1 OM 12 - I AM 12 ==
I OM 12 - 1 BM 12 == 1 OB 12 == R2. Hence, M lies
on the straight line perpendicular to OA (see Problem t of Sec. 2).
9. The condition defining the set of points M
is equivalent to the condition' AM 12 -k2 , BMI2=
0, that is, this is a circle (see Problem 7 in
Sec. 2). This circle is called A pollonius' circle;
obviously, its centre lies on the straight line AB.
10. Since M B is the bisector of the angle A Me,

+
+

IAMI
I Me I

IABI
TBCT·

+ +

+

.

Consequently, the bisector of
the exterior angle with respect to the angle A MC
intersects the line A C at a constant point K:


216

IAKI
IABI
TKCT =~ , and

.

the sought-for set of points
M is the arc of the circle constructed on B K as
.diameter and enclosed between the straight lines
perpendicular to the line segment A C and passing
through the points A and c.
t t. Let 0 1 and O 2 be the centres of the given
circles, 'I and r 2 their radii, M a point of the
desired set, MAl and MAt tangents. By hypothe2
sis, I MA11=k I MAt I. Consequently, I MOl 1 _
k 2 I MOt 12 = r~ - k 2 rl . Hence (see Problem 6 of
Sec. 2), for k =1= 1, the sought-for set of points M
is a circle with centre on the straight line
2,
while for k = 1, the desired set is a straight line
perpendicular to 010t.
12. Let (Fig. 15) K and L be the points of
intersection of the tangent to the second circle

°.0

/.,

Ii'

Fig. i5
passing through D and the tangents to the first
circle passing through B and A, and M and N two
other points. It is obvious that LDKB == L CMA
(either of these angles is equal to one-half of the
difference between the angles corresponding to the
arcs A B and CD) • Therefore (in the figure)
L.LMN
LLKN = 180°. Consequently, KLMN

+


217

is an inscribed quadrilateral. Further, we have

I DK I sin L DBK sin ~ -- AB
. The ratios
I KB I -sin L BDK- · 1 -DC
SID"2 ......

of the lengths of the tangents drawn through the
points L, M, and N are found in a similar way. All
these ratios are equal; hence, the centre of the
circle circumscribed about KLMN lies on the
straight line passing through the centres of the
given circles (see Problem 6 in Sec. 2).
13. Expressing the distances from the vertices
of the triangle to the points of tangency, check the
fulfillment of the conditions of Problem 3 in
Sec. 2.
14. Let I AMI I : I BMJ...I : I eM I I = p : q : r,
Then the set of points M such that(r2 - q2)IAM 11
(p2 _ r2) I BM 13
(q2 - p2) I CM 12 = 0 is a
straight line passing through M l' M 2' and the
centre of the circle circumscribed about the triangle
ABC (see Problem 7 in Sec. 2).
15. Points M I and M 2 belong to the set of points
Mfofwhich5 I MA pI - 81 MB pa
31 Me 12 =
O. This set is a straight line, and, obviously, the
centre of the circumscribed circle satisfies the condition that defines this set (see Problem 7 of Sec. 2).
16. Let I AA I I = a, I BB I , = b, 1 cc, I = c,
1 AtB I 1 = e, I BtCI , = y, ICtAl' = s, Then
2
I AB l 1 = a2 + x'l, IB1 CI2 = c2
y2 and 80 forth.
Now, it is easy to check the conditions of Problem 3,
Sec. 2.
17. Let I AD I = x, I BD I = y, I CD I = z,
I AB 1= a. Let A 2 , s ; C 2 denote the points of
tangency of the circles inscribed in the triangles
BCD, CAD, ABD, respectively, with sides BC,
CA, A B. The perpendiculars drawn through the
points A I' s., C t to the sides BC, CA, and AB coincide with those drawn to the same sides at the
a+y-z
points At, s ; Ct· But J BAt I =
2

+

+

+

+


218

, A 2C I

=

a+z-y

2
; I A C2 I, I CsB I, I A B'}, I,
I B"C I are found in a similar way. Now, it is

easy to check the conditions of Problem 3, Sec. 2.
18. Apply the conditions of Problem 3 in
Sec. 2, taking the centres of the circles as the points
A, B, and C, and each one of the two intersection
points of the circles as the points At, B t , C 1 (At is
one of the points of intersection of the circles with
.centres Band C, and so on).
19. Take the third circle with diameter BC. The
altitudes of the triangle drawn from the vertices
Band C are common chords of the first and third,
and also the second and third circles. Consequently
(see Problem fS in Sec. 2), the common chord of the
given circles also passes through the intersection
20. Let 0 denote the centre of the given circle,
R its radius, MC a tangent to the circle. We have
2
2_1 MN 1 = I MO I' - I MB 1·1 MA 1=
I MO 1
2
I MO 1 - 1 MC II = R2, that is, the point M

lies on the straight line perpendicular to the
straight line ON (see Problem 1 in Sec. 2). It can
be easily shown that all the points of this line
belong to the set.
21. Let 0 denote the centre of the circle, r the
radius of the circle, I OA I = a, Be a chord passing
through A, and M the point of intersection of the
tangents. Then
2
I OM a = I BM 1
r2 ,

r

+

IAMII= IBMJI- 1IBCII+(

+

~ IBCI-IBAI

y=

IBMII-IBCI·IBAI
IBAI'=
IBMI 2-IBA1·1ACI = IBMII-r 2+al •
Thus, I OM 12 - I AM 12 = 2r 2 - a2 , that is (see
Problem 1 of Sec. 2) the required set of points is
a straight line perpendicular to OA. This line is
called the polar of the point A with respect to the
given circle.

219

22. Show that if M I and M 2 are two distinct
points belonging to the set, then any point M of
the segment of the straight line M1M 2 enclosed
inside the triangle also belongs to this set. To this
end, let us denote by XI' YI, and %1 the distances
from M. to the sides of the triangle, and by %2' Y2'
%, the distances from M t. Then we can express the
distances x, Y, % from M to the sides of the triangle
in terms of those quantities and the distances
between M I , u ; M. For instance, if I M1M 1=
k I M IM 2 I and the directions of M 1M and M.M"
coincide,
then x = (1 - k) %1
kXt,
Y=
(1 - k) YI
ky", z = (1 - k) %1
kZt. Hence, it
follows that if the equality is true for three noncollinear points inside the triangle, then it is true
for all the points of the triangle.
Remark. The statement of the problem remains
true for an arbitrary convex polygon. Moreover,
we may consider all the points in the plane, but the
distances to the straight line from the points situated on opposite sides of the line must be taken with
opposite signs.
23. For the distances z, y, % to be the sides of
a triangle, it is necessary and sufficient that the
inequalities x < Y
I, Y < z
x, % < X
Y he
fulfilled. But the set of points for which, for inz is a line segment with the end
stance, x = y
points lying at the feet of the angle bisectors (at the
foot of the angle bisector two distances are equal,
the third being equal to zero; consequently, the
equality is true; and from the preceding problem
it follows that this equality is true for all points
of the line segment).
A nnaer: the sought-for locus consists of points
situated inside the triangle with vertices at the
feet of the angle bisectors.
24. Since the perpendiculars from A 2' B 2' and
C 2 on BtC t, CtA t, and AIB t , respectively, are
concurrent, the perpendiculars from A l ' B t , and
CIon B 2Ci' etA 2' and A "B 2' respectively, are
also concurrent (see Problem It of Sec. 2).

+

+

+

+

+

+

+


220

25. Let al and al denote the distances from A
to the straight lines It and 13 , respectively, bl and
b t the distances from B to the straight lines Is
and I., respectively, Ct and CI the distances from C
to the straight lines 1 and It, respectively, x, y,
and z the distances from Atl B., and Ct to l, respectively. For the perpendiculars drawn respectively from A, B, and C on B.C., C.A t , and AIB I ,
it is necessary and sufficient that the following
equality be true (see Problem 3 of Sec. 2): I AB I 12 _
1
2
2
I,BIC 1 1CA l I2_ 1AlB 1 I BC 1 1 - 1CtA 12 = 0
or (a1+ y2)_ (ci+ 1I2)+ (cl+x2 ) - (b~+Z2)+ (bi+
z2)- (al + Z2) = 0 which.. leads to the condition
al- 41 bl- bl C~ - cl= 0, independent of e, y, z.
26. It suffices to check the fulfillment of the
condition (see Problem 3 of Sec. 2) I AB t II 2
I BIC 12 + I CAl pa - 1 A 2B 1
I Bel 12 I CsA II = O. Note that the triangles BBIC I and
AA,CI are similar, hence, I AC I 1·1 CIB I 1=
I BC I 1·1 CtA I ; in addition, LACaB II= LBCtA lI,

+

+

+

+

+

fogt~q~)~ ~i c~:~1 ~IL~~,r

By ~j3~~lIth;

corresponding equalities for I CAl 12 - I AC I 12
and I BC I 12 - I CB 2 12 and adding them together,
we see that the sum of the difference in the first
parentheses yields zero (apply the conditions of
Problem 3 of See. 2 to the triangles ABC and
AIBICI; we get zero since the altitudes intersect
at one point). It is easy to prove that AA I' BB l t
and CC, pass through the centre of the circle circumscribed about ABC, that is, the sum of the
differences in the second parentheses is also zero.
32. Through K and L, draw straight lines
parallel to Be to intersect the median AD at points
Nand S.Let I AD I = 3a, I MN I = za, I MS 1 =
.
ILS!
IASI
ILSI
fMSI
ya. Since

have

INKI

=

IANI'

IASI
IMSI
IANI = IMNI'

INKI = IMNI ' we

(2+y) a
(2-z)a

=.L,
z

y=


1~x'
.

IS

IM~I

The equality

·

1

equrva ent to

221

t

IMNI =

I~LI + lipi

=,
1

t

1

IMDI

IMSI -

1 r- t = - -I t . Substitut iing y= -xu st.itut

we get
ax
ay
a
i-x'
a true equality.
34. Let 0 be the point of intersection of the
diagonals A C and BD; taking advantage of the
similarity of the appropriate triangles, we get

lOKI

1OCT=

lOKI
lOBI

lOBI

-lOCI=:

IOAI 10MI IOMI
10DI - 10AI = IODI '

which was to be proved.
35. Let F and D denote the points of intersection
of EN and EM with AB and BC, respectively.
Prove that the triangles A FNand M DC are similar. Using the similarity of various triangles and
equality of the opposite sides of the parallelogram,
we have:

I ED I

TFAT=

INFI

INFI

IFBI

'!FAT= TFBT · TFAT ==
I BD I
I DM I

I DC I

-TFET===

IBDI

I DM I X

I BD I

I DC I

I DM I

I BD I

II~~ II ,that is, the triangle A FN is similar to

the triangle M DC.
36. The statement of the problem becomes
obvious from the following two facts:
(1) If, on the sides of the quadrilateral ABeD,
points K, L, M, and N are taken so that the sides
AB, BC, CD, and DA are divided by them in the
I BK I
I CM I
I BL I
same
ratio
( TKAT= I MD I - I LC I -

II ~~: ) • then the line segments KM and LN are
also divided in the same ratio by the point P of
their intersection.

222


Indeed, from the fact that the straight lines KL
and N M are parallel to the diagonal A Cit follows
IKPI
IKLI
IKLI
IACI
18KI
that IPMI = lNMI =: IACI - INMI = IBAI X
IADI
IBKI
IBAI
IBKI
INDI =·=lBAT- IKAI ==lK'At2) If, on the sides AB and CD of the quadrilateral, points K I and K, M 1 and M are taken
IK1KI
IMIMI
t
so that IABI
ICDI
m ' IAKII = IKBI,

I DM I I

=

CM, then the area of the quadrilateral

K lKM M 1 is...!.. of the area of the quadrilateral
m

ABCD. Indeed,

IMIDI

ICDt

IRKI

= lBAI

SACD

IRKI )
( t-lBAT"

SAKCM 1 =

· ·1
S nm ar I y, S K1KMM 1
S

_
K1KMM 1 -

IRKI
IBAI

SBKC=

SABC, SAAl t D =

SACD.

Consequently,

IAKI

SABCD=!BAT'SABCD.

IK1KI S AKCM 1 •
IAKI
IK1KI
_ 1
IABI SABCDS.

=

Th us,

m

37_ Let K be the midpoint of DB, L that of
AC, SANM = SCNM (since I AL I = I LC I). In
similar fashion, S BNM = SDNM' whence there
follows the statement of the problem.
38. If M is the midpoint of DC, N that of BC,
K and L are the points of intersection of DN
with AM and AB, respectively, then
IDMI

t

l"A'LI=T'

that
4

quentlY,SADK =

5

.

IS,

IAKI

S ADM =

4

=5
4

1

:~~I-

t AMI;

1

5" - T S = 5

conseS (8 the

223


parallelogram). Thus, the area of

area of the

~

the sought-for figure is S - 4S ADK =

S.

39. Let Q, N, and M be the midpoints of AD,
Be, and DC; K, P, and R the points of intersection of DN and AM, QC and DN, and QC and
respectively.

AM,

IDPI = IPNI,

I DK I

Then

IQPI = IPCI, IQRI =

IRPI
IKPI
IQPI · IDPI

SRPQ
SQPD

1

15

2
= 5" I DN I ,

S

1

== 3"

+

IQCI,

1

1

5 == 15 '

X

S

8= 120

•
Consequently, from the quadrilateral considered
in the preceding problem, four triangles, each

SRPK =

X

having an area of

1~O

are thus cut off, the area

"
S
48
8
of hed esire d octagon bei
t
emg '"'5- 120 =6 ·
40. Let the straight line He intersect A Band
LM at points T and N, respectively, the straight
line A L intersect ED at a point K, and the straight
line BM intersect PG at a point P. We have:

==

==

SACDE= SACHK
SATNL' S BCFG
S BCHP=
SBMNT; thus, SACDE
SBCFG
SABMLo

+

==

41. Let Q denote the area of the pentagon,
and Sa the areas of the triangles adjoining
one of the lateral sides, the smaller base, and the
other lateral side, respectively; x the area of the
triangle enclosed between the triangles of areas 81
and ss, and y the area of the triangle enclosed
between the triangles having areas 's and 830 Then

SI' Ss,

81

+ +
X

and, thus,
82

82
81

+ Q ~ 81

1
+ y + == 2" (x + y + + Q)
+ x + + +Y+ = +y +
+ + = Q.

==

82

82

Sa

8S
Sa

82

S2

8a

2:

224


42. If S is the area of the parallelogram, then
'. ,
1
SABK + SKCD = 2" S.
On the other hand,
1
+ 8 K CD = "2 S,

hence, SABK =
Analogously, SAKD = SKCP; adding together the last tW9 equalities, we get: SABKD =
SDBC

= SEKC

SEKCo

SCEKF·

43. We have:

IACll
IC1BI =

+IACI'~~llsinLACCl
1

"2 ICC11·ICBI sin

L 0lCB

=

SACCa
SeCtB

=

IAC sinLACC)
I Be I • sin L C1 CB •

Having obtained similar equalities for the ratios
IRA.I
ICBll
..
IAICI and IB1AI and mult iplying them, we
get the required statement.
44. Let us show that if the straight lines intersect at the same point (let M denote this point),
then R* = 1 (and consequently, R = 1; see
Problem 43, Sec. 2). By the law of sines for the
.
sin LACC t
I AM I
triangle AMC we have: sin LAtAC
t MC I ·
Writing out similar equalities for the triangles
AMB and BMC and multiplying them, we get the
required assertion. Conversely: if R = t, and all
.the points AI' B l , C1 lie on the sides of the triangle
(or only one of them), then, drawing the straight
lines AA 1 and BB 1 , we denote the point of their
intersection by M 1; let the straight line eM 1 intersect A B at a point C I. Taking into consideration the conditions of the problem and that the
necessary condition R = 1 is proved, we have:
I AC 1 I
I ACIII ' bot h 0 f t he noi
d
, C1B I
I C"B
e points C1 an C,
lying .either on the line segment A B or outside it.
Consequently, C1 and C'l coincide.


225

45. Let AI' Bit CI be collinear. Through C, we
draw a straight line parallel to A B and denote the
point of its intersection with the straight line
AIB I by M. From the similarity of appropriate
.
I
1BA I I
I Bet I
I CB I I
trtang es, we get: I AIC I
I eM I ' I RIA I

I CM
I A ellI R epI ·
acing t h e correspon dine ratios i
ing ratios In
the expression for R (see Problem 43 of Sec. 2)
with the aid of those equalities, we get: R = 1.
The converse is proved much in the same way as
it was done in the preceding problem (we draw
the straight line BIA}., denote the point of its
intersection with AB by C", and so forth).
46. Check the following: if for the given straight
lines R* = 1, then for the symmetric lines the
same is true. If the straight line passing, say,
through the vertex A intersects the side BC, then
the line symmetric to it with respect to the bisector
of the angle will also intersect the side BC (see
Problems 43 and 44 in Sec. 2).
47. If A 0' B o, Co are the midpoints of the line
segments AO, BO, CO, respectively, then the
constructed straight lines turn out to be symmetric
to the lines AoO, BoO, CoO with respect to the
angle bisectors of the triangle A oBoC0 (see the
preceding problem).
48. (a) Let the straight line BM intersect A C
at a point H', and the line CK intersect AB at a
point C'. Through M, we draw a straight line
parallel to A C and denote by P and Q the points
of its intersection with AB and BC, respectively.
.
IAB'I
IPMI
.
Obviously, I B'C I -:.=...: I MQ I · Drawing through
K a straight line parallel to A B and denoting by
E and F the points of its intersection with CA and
.
IBC'I
IFKI
CB, respectively, we have: I C' A I - I KE t • We
earry out a similar construction for the point L.
Replacing the ratios entering the expression for R
15-01557

226

Problems in Plane GeometrY

(see Problem 43 of Sec. 2) with the aid of that
equality we take into account that for each line
segment in the numerator there is an equal segment
in the denominator,for instance: I PM 1=1 KE I.
(b) Let, for the sake of definiteness, the line I
intersect the line segments CoA, CA 0 and form an
acute angle q> with OK. The straight line AIL
S

divides the line segment M K in the ratio SLMA.
LKA 1

(starting from the point M). The ratios in which
the sides KL and LM of the triangle KLM are
divided can be found in a similar way. We have
to prove that there holds the equality R = t
(see Problem 43, Sec. 2). Let us replace the ratios
of the line segments by the ratio of the areas of
the corresponding triangles. Then R will contain
SLMA 1 in the numerator and SKMClin the denom-

s~n C where A
sm A,
and C are angles of the triangle ABC. Obviously
SBoOA o
sin C
-'""'"""---- - - In addition, L AIBoAo =
S BoOCo . - sin A •
inator.

Prove

that

SSLMA.
KMC 1

+ L A IBoCo=90° -

+ep

L B
(this fol2
lows from the fact that the circle of diameter AD
passes through B o, Co and At) and L. BoAIO =
L CoBoA o

L BoAO =

~ C and L
(90° -

In similar fashion L BOCIO=

CLBoCo= (90° -

L B)
2

+ L CIOL=

L B )+(180°- L C- L BOOC I)
2

~B +( L
L A -

L A
2

=

90°_

BoOAI- L C)=90o- L Bj2+(1800-

L C -

cp) =

900

+

L B/2 -

cp,

Le.,

227


denote

r

C

C

o

IPoA11.IBoAol
IBoC11·IBoCol -

sin TOcOS 2
sine
. A
A:;:; sin A

sin

the radius of

lOLl = lOKI = IOMI

Let

S
T· cO 2
the

inscribed circle

= a. We have:

SLMA 1

__

SLOM+S LOMA t

S KMC I

-

S KOM

Z

+S KOMC

a
rr AoOB o + -,: S A,OBoA

a

-

a

l

-,:I

I

S

S

CoOB o

I

+-;- S C.OBoC
a

t

.!!- S AoOB, +(8 AIB.A t - 8 AoOB o)
r
!!..S CoOB. +(SCoBoC. -SCoOB o)
r

(-7 -1) 8

A.OB.

+8A.B.A,

- (a -1 ) SCoOB o +SC.BoC
-r

sin C
sin A •

t

(The latter of the equalities follows "from the
fact that S A.OB.

S A.B,A t

SC.OB,

SC.BOCI

sin C )
sin A •
In similar fashion, we single out in the numerator
and denominator of the expression for R, two more
pairs of magnitudes whose ratios are equal to
sin A
sinB
.~
--:---B and --:--c ,respectively. Hence, R = 1. It
sin
sin
remains only to prove that the number of points
t 5*

228


of intersection of the straight lines LA l , KC. t and
MB I with the line segments KM, ML, and LX,
respectively, is odd.
49. Consider the triangle ACE through whose
vertices the straight lines AD, CF, and EB are
drawn. The sines of the angles formed by these
lines with the sides of the triangle A CE are proportional to the chords they are based on; consequently, the condition R = 1 (see Problem 44
of Sec. 2) is equivalent to the condition given in
the problem.
50. Find out whether the equality R = 1 is
fulfilled (in Item (b) use the result obtained in
Problem 234 of Sec. 1) and all the three points lie
on the extensions of the sides of the triangle. Thus,
our statement follows from Menelaus' theorem
(see Problem 45 of Sec. 2)_
51. By the property of the secants drawn from
an exterior point to a circle, or by the property
of the segments of the chords of a circle passing
through the same point, we have: I BC1 I· , BCI I =
I BAli-I BAIl, I CB11-1 CBgl=1 CAII·t CAli,
I ABII·I ABII = I ACI'·I ACt ,. Now, It is easy
to check that if the assertion in Ceva' s theorem
(the equality R = 1) is true for the points
AI' B l , CIt then it is also true for the fOints A I'
B I , Ct " It follows from the statement 0 the problem that either all the three points A It B I' C I lie
on the corresponding sides of the triangle or only
one of them (see Problem 44 of Sec. 2).
52. Writing out the equality R = 1 (according
to Ceva's and Menelaus' theorems-see problems
44 and 45 in Sec. 2) for the points AI, B I , C 1 ;
AI, B I , C.; At, B s' Cl ; and AI' B l • CI , we get that
for the points A It B I' C s the equality R = 1 is also
true. Now, it remains only to prove that either
all the three points A It B I' and CI lie on the
extensions of the sides of the triangle (that is the
case when the points A lt B l , C l are found on the
sides of the triangle) or only one lies on the extension (if only one of the points AI, Bland C 1 is on

229


the sides of the triangle) and use Menelaus' theorem
(see Problem 45 of Sec. 2).
53. Make use of Menelaus' theorem (see Problem
45 of Sec. 2). As the vertices of the given triangle,
take the midpoints of the sides of the triangle ABC
on whose sides and their extensions the points under
consideration lie.
54. If a is the length of the side of the pentagon
MKLNP, b the length of the side of the pentagon
with one side on A B, c the length of the side of
the pentagon whose one side is on A C, then
I BA I I =!!:.. I ACtl = !.- • I CB t I _..!. M II CIB I
b' I BIA I e ' I Ale I - a· u
tiplying these equalities, we find R = 1 and then
use Ceva's theorem (Problem 44 of Sec. 2).
55. Check to see that the points AI' A S, A I and
B I' B , B 3 are found either on the sides 01 the
triangle 0 10.°8 (0 1 , 0 1 , 0 3 centres of the circles)
or on their extensions, and the ratio of the distances
from each of these points to the corresponding
vertices of the triangle 0 10 103 is equal to the
ratio of the radii of the corresponding circles.
Further, make use of Menelaus' theorem (see
Problem 45 of Sec. 2) for each of these three points.
56. The statement of the problem follows from
Problems 43 and 44 of Sec. 2.
.
sin L B 1AA 2
58. Make use of the equality
.
A AC =
Sin L I t

IACll IB 1A2 1
Obtaining similar equalities for
IAB!t lA t Cl l
the other angles and multiplying them, we get
our statement on the strength of the resul ts of
Problems 43 and 44 of Sec. 2.
59. We apply Menelaus' theorem to the triangles ABD, BDC, and DCA (Problem 45· in Sec. 2,
AL
BQ DP
BM CR
LB· QD· PA
-1, Me· RD X
DQ
AP
DR
CN
QB =-1, PD • RC· NA =-1 (L,M, and N

Remark):

=


230

the points of intersection of AB and PQ, Be and
QR, AC and PR, respectively). Multiplying these

· ·
t CN
AL
BM
i
equa 1Itle$, we ge: NA· LB ·.MC == - , th at
is, the points L, M and N are collinear.
60. Consider the coordinate system whose axes
are the given lines (this is the affine system of
coordinates). The equation of a straight line in
this system, in the usual fashion, has the form
ax + by
c = o. We shall first prove the necessary condition. Let the point N have coordinates
(u, v) and the point M the coordinates (Au, Av).
The equations of the straight lines AlB l' A ,B?,
A 3B a, A.B. have the form: y - v = kl (x - U),
Y - v = k s (x - u), y - AV = k a (x - AU), y AV = It. (z - AU), respectively. Then the points
A l' A I' A s, A. situated on the z-axis have, respectively, the coordinates on this axis: u V,

+

!

'''"I

;4

U : . v, )..u - ;. V, )..u V, while the points
Bit B I , B a, B. situated on the y-axis have the
coordinates v - k 1 u , V - k1u, Av - k,Au, AVk,Au, respectively. Now, it is easy to check the
equality given in the hypothesis. Sufficiency, in
usual fashion, can be proved by contradiction.
61. In Items (a) and (c), make use of Ceva's
and Menelaus' theorems (Problems 44 and 45 of
Sec. 2, Remark). In Item (b), in addition, use the
result of the preceding problem; here, it is convenient, as in Problem 60, to consider the affine
coordinate system whose axes are the straight
lines AB and AC, and the points Band C have
the coordinates (0, t) and (1, 0).
62. Let S denote the point of intersection of the
straight lines AtM, BIL, and c.s, Applying
Menelaus' theorem (Prob em 45 in Sec. 2, Remark)
to the triangles SMK, SKL, and SLM, we get

KL 1
LIM

•

MAl. SCI =-1 LM l
AlB
ClK
'MIK

•

KC l
CIS

•

SB 1
B.L

=

231

MK I
LB I • SAl - -1
K1L
B1S
AIM .
these equalities we get:
-1

,

KL 1
LIM

MK 1 - - 1
(1)
KIL ·
Equality (1) is a necessary and sufficient condition
for the lines AIM, B1L, and CIK to intersect at
•

LM I
MIX

Multiplying

•

a point. The necessity has been already proved.
The sufficiency is proved, as usually, by contradiction. (Let us denote by S' the point of intersection of AIM and BtL, draw S'C I , denote by K '
the point of its intersection with the given straight
line, and prove that K and X' coincide.) Since the
equality (1) goes over into itself with K, L, M
replaced by K l , L 1 , M h respectively, and vice
versa, the assertion of the problem has been
proved.
63. Applying Ceva's theorem (Problem 44* in
Sec. 2, Remark) to the triangles ABD, BDC and
CDA, we get:
AP BF DE
BQ CG
DF
CR
PB • FD· EA =1, QC· GD • FB =1, RA X

:: ·gg

= 1. Multiplying these equalities, we

get: : ; .

gg · i~ =

1, that is, the straight

lines AQ, BR and CP intersect at a point. Let us
denote it by N. Let T be the point of intersection of
PG and DN. By Menelaus' theorem we have
DT NP CG
DT
PC
TN· PC • GD =-1, whence TN = - NP X
GD
CP GD
AE
BF
CG ==- PN • CG·
If
ED =ct,
FD =p,
CG
GD =y,

then

_ BA. RC =
PB AR

AP
PB

a

=If'
a+~ . ..!..
P

CR
RA

CP
<x' PN

y

=a-'

cs
NP

=

=-(1+ eN)=
NP

232


a.~+py+ya

DT _a~+~y+ya. Th
us, TN apy
•
e
line segment DN is divided in the same ratio by
the other straight lines.
64. Let us first consider the limiting case when
the point N is found at infinity; then the straight
lines AN, BN, and CN are parallel to the straight
line I. Let the distances from the points A, B, and
C to the line l be equal to a, b, and c. (For convenience, let us assume that A, B, and C are on
the same side of l.) The straight lines parallel to l
and passing through A, B, and C intersect the
straight lines BtC tt CIA tt and AIB I at points
At, B t , CI , respectively. It is easy to see that
I Aiel I a c I BlAt I b a. I CtB I I
I CtB t I = c b ' ,A 2el I = a + c ' I B 2 A 1 I =

ap

:t ~.

+
+

·

Th

+

Multiplying these equalities, we make
sure that the statement of Menelaus' theorem
(Problem 45 of See. 2) is fulfilled (it is necessary
also to make sure that an odd number of points
from among At, B 2' C 2 are found on the extensions
of the sides ot the triangle AIBIC I ) . Hence, the
points A I' B 2' C 2 are collinear.
The general case can be reduced to the considered one if, for instance, the given arrangement of
the triangles is projected from a point in space on
another plane. In choosing this point, we should
get that the symmetry of the triangles is not violated, and the point N tends to infinity. It is also
possible not to resort to spatial examinations. Let
us introduce a coordinate system with the straight
line l as the z-axis and the origin at N. We carry
out the transformation x' = 1/x, y' = ylx. As a
result of this transformation, the points of the
x-axis (y = 0) go into the straight line y' = 0;
the points symmetric about the x-axis go into the
points symmetric with respect to the line y' = 0;
also straight lines go into straight lines; straight
lines passing through the origill go into straight


233

lines parallel to the line y' = 0 (this transformation is, in essence, the projection mentioned above).
When this transformation is carried out, we get
the arrangement we considered.
65. We assume the given lines to be parallel.
This can be achieved by projecting or transforming
the coordinates (see the solution of Problem 64
A,

As

A
J

of Sec. 2). Apply Menelaus' theorem (Problem 45
of Sec. 2) to the triangle AIA.M (Fig. 16, N' K'
is parallel to the given straight lines). We have
(AILI fA.K, IMNI
I AlAs I IAeAl1
t LA. I I KM I I N Al I I A 4A. l • I K' M I X
IMN'I
IAIAsl
IMN'I
IA I A21
t· A.A I I = I K' MI' I A.A. I
I AliAI f =
I.AtA11 IMAil (A,Asl
IAIMI
I MAli I
~f AIM I • I AliA, I · I AliA! 1 I A 2M I · I A&M I X
IA~I

I MAl I

.

1. Thus, the points L, N, and K are
~11inear. According to the Remark to Problems 44
~ 45 of Sec. 2, we could consider the ratios
"'JIlL
·
d
I AlL I
:~A.
and
others
instea
of
I LA, I
.~

others. In this case the product of the appropriate ratios is equal to (-1).

234


67. The desired locus consists of two straight
lines passing through the point symmetric to the
point A with respect to the straight line land
forming angles of 60° with I.
68. The required set is the arc B C of the circle
circumscribed about the triangle ABC corresponding to a central angle of 120°.
69. If N is the point of intersection of the
.
.
I CN I I PC I
straight Ilnes PQ and AB, then I AN I = I AQ I =
~~ :' that is, N is a fixed point. The required
set is a circle with diameter CN. If now M is a
fixed point, then D lies on the straight line parallel
to the line M N and passing through a fixed point L
. . .
I AL I
I AN I
on the straight line AB such that I LB I == I eN I'
L being arranged relative to the line segment A B
in the same manner as N with respect to the line
segment AC.
70. Let cp denote the angle between BD and A C;

SAPK=~ I AK 1·1 PD l sin e, SBPC= ~ I BP I X
I DC Isincp= ~ I BP 1·1 AD Isinql. Since SAPX=
SBPC,

IAKI·IPDI=IBPI·IADI, or

:~~ I

triangle

IAKI
TADTX

=1, but by Menelaus' theorem for the
BDK (see Problem 45 of Sec. 2).

r AX I . I DP I . I BM I = 1
I AD I I PB I I MK I

(M the point of intersection of A P and B K), consequently I BM I = I MK I, that is, the required
locus is the midline of the triangle ABC parallel
to the side A C (if the points P and K are taken
on the straight lines AC and BD, then we get a
straight line parallel to the side A C passing through
the midpoints of the line segments AB and Be).

~swerSt

Hints, Solutions

235

71. Let C denote the vertex of the given angle,
and p its size. We drop perpendiculars OK and OL
from 0 on the sides of the angle (Fig. 17t a).
A circle can be circumscribed about the quadrilateral OKAM. Consequently, LKMO = LKAO.
Analogously, LOML = LOBL. Hence, LKML =
L"KAO+ LOBL = a,+ p, that is, M lies on

c

c

Fig. 17
an arc of the circle paSSin, through K and Land
containing the angle a,
,all the points of this
arc belonging to the set. I a ~ Ii, then there are
noother points in the set. And if a, > p, then added
to the set are points M located on the other side of
the straight line KL for whieh L KML = a, - P
(Fig. t 7, b). In this case, the set of points is a pair
ef arcs whose end points are determined by the
limiting positions of the angle A OB. If the rays
c)!:, the fixed angle p and movable angle a are extended, and instead of the angles, the pairs of
straight lines are considered, then the desired
let is a pair of circles (containing both arcs mentioned above) .
72. Consider the quadrilateral DEPM in which
-4DEM = LDPM = 90°, consequently, this is
~n inscribed quadrilateral. Hence,
LDME =
£ DPE = 45°. The required locus is the straight

+

line DC.

73. Consider the case when the point B lies
inside the given angle. First of all we note that

236


all possible triangles BCD (Fig. t8) are similar
since LBCD == LBAD, L.BDC == LBAC. Therefore, if N is the midpoint of CD, then the angles
BNC and BND are constant. Let us circumscribe
a circle about the triangle B N C and let K be the
second point of intersection of this circle and A C.

Fig. 18
Since LBKA = 1800 - LBNG, K is a fixed
point. Analogously, also fixed is L, the second
point of intersection of the circle circumscribed
about the triangle BND and the straight line AD.
We have: LLNK = LLNB
LBNK == 1800 LBDA
LBCK == 180°, that is, N lies on the
straight line LK. The set of points N is the line
segment LX t and the locus of centres of mass of
the triangle A CD is the line segment parallel to
LK dividing A K in the ratio 2 : 1 (obtained with
the aid of ahomothetic transformation with centre
at A and the ratio of similitude equal to 2/3).
74. If 0 is the vertex of the angle, ABCD is a
rectangle (A fixed) , then the points A, B , C, D and

+

+


237

o lie

on the same circle. Consequently, L COA =.
000 , that is, the point C lies on the straight line
perpendicular to OA and passing through O.
75. Note that all the triangles ABC obtained
ar&. similar. Consequently, if we take in each
triangle a point K dividing the side BC in the
same ratio, then, since L. A KC remains unchanged,
the point K describes a circle. Hence, the point M
dividing A K in a constant ratio also describes a
circle which is obtained from the first circle by a
homothetic transformation with centre at A and
the ratio of similitude k = I A M 1/1 A K I. This
reasoning is used in all the items: (a), (b), and (c).
76. Let K denote the midpoint of AS, and M
the foot of the perpendicular dropped from K OD
AC. All the triangles AKM are similar (by two
congruent angles), consequently, all the triangles
ABM are similar. Now, it is easy to get that the
desired locus is a circle with a chord Be, the angles
based on this chord being equal either to the angle
AMB or to its complementary angle. (The minor
are of this circle lies on the same side of Be as
the minor arc of the original circle.)
77. If M, N, L, and K are the given points
(M and N lie on opposite sides of the rectangle as
~

P~------40

Fig. 19
do Land K), P is the midpoint of M N, Q the midpoint of KL, 0 the intersection point of the diagonals of the rectangle (Fig. 19), then L POQ =

238


900 • Consequently, the desired locus is the circle
constructed on PQ as diameter.
78. Let Rand r denote the radii of the given
circles (R :>- r), D the point of tangency of the
chord Be and the smaller circle. Let K and L be the
points of intersection of the chords A C and A B
with the smaller circle, and, finally, let D be the
centre of the circle inscribed in the triangle ABC.
Since the angular measures of the arcs A K and
AC are equal, I AK I = rx; I AC I == Rx; hence,
2
we get I DC 1 == I AC 1·1 CX I = (R - r) Rx".
Similarly, I AB I = Ry, I DB 12 == (R - r) R y2;

I CD I

x

I AC I

.

consequently, 'DB I
y =TABT' that IS, AD
is the bisector of the angle BAC. Further, we have:
I AD l
I AC I
Rx
.. /
R
I OD I
I CD I
Y(R-r)Rx
V n-:«:
Thus, the desired locus is a circle touching internally the two given circles at the same point A
with radius
'AD,

rYR

p=rTADT;:::: YR+YR-r •
79. Let 0 1 and Os denote the centres of the
given circles, the straight line DID" intersect the
circles at points A, B, C, and D (in succession).
Consider two cases:
(a) The rectangle KLMN is arranged so that
the opposite vertices K and M lie on one circle
while Land N on the other. In this case, if P is
the point of intersection of the diagonals
2
(Fig. 20, a), then I DIP 1 - I 02 P I" = (I 0lK 12 _
2
2
I KP II) - (I O,L I" - I LP 1) = I OtK 1 lOlL 12 = RY - Rl, where R 1 and R" are the
radii of the circles, that is, the point P lies on a
common chord of the circles; the midpoint of the
common chord and its end points are excluded,
since in this case the rectangle degenerates.
(b) Two neighbouring vertices of the rectangle
KLMN lie on one circle, and two others on the

Answers, Hints, SOlutions

239

240


other circle. Since the perpendiculars from 0 1 on
KN -,and from 0 1 on LM must bisect them, the
straight line 0 10 1 is the axis of symmetry for the
rectangle KLM N.
Let R t be less than R 1 and the radius 02L form
an angle cp with the line of centres. We draw
through L a straight line parallel to OlD". This
line intersects the circle 0 1 at two points K and
K I' and to the point L there will correspon two
rectangles: K 1LMN1 and K 2LMN s (Fig. 20, b).
With q> varying from 0 to n/2, the angle", formed
by the radius 0IK I and the ray DID' varies from 0
to a certain value '1'0. With a further change in cp
(from n/2 to n}, , decreases from 'i'o to O. Meanwhile, the centres of rectangles K1LM N I will
trace a line segment from the midpoint of CD
to the midpoint of BC excluding the extreme points
and the point of intersection of this line segment
with the common chord. Analogously, the centres
of rectangles K 2LM N I will fill in the interval with
end points at the midpoints of AB and AD (the
end points of the interval are not contained in the
locus).
If three vertices of the rectangle and, hence,
the fourth one lie on a circle, then the centre of
the rectangle coincides with the centre of the
corresponding circle.
Thus, the locus is the union of three intervals:
the end points of the first interval-the midpoints
of AB and AD, respectively, the end points of the
second interval-the midpoints of Be and CD,
the end points of the third interval-the points of
intersection of the circles, the midpoint of the
common chord being excluded.
so. If Band C are the first and second points
of reflection, 0 the centre, then BO is the bisector
of the angle CBA. The path of the ball is symmetric with respect to the diameter containing C,
therefore A lies on this diameter. If L BCD =
L CBO = cp, then L ABO = cp, L BOA = 2q>; applying the law of sines to the triangle ABO

J

Answerst Hints. S olutions
(180

I = R,

lOA

I=

241

a), we get: --!!--3 =~,
81n cp
SID = ~. and for a > 3' we can
iind cp.
A nsurer: points situated outside the circle of
radius RIa centred at the centre of the billiards.
~t. The required locus are two straight lines
perpendicular to the given lines.
82. If the line AB is not parallel to I, then there
are two circles passing through A and Band
touching I. Let 0 1 and 0, denote their centres.
The sought-for locus is the straight line 0 10 1
ex.eluding the interval (010,.). If AB is parallel to
I, ;then the desired locus consists of one ray per~endicular to I.
j,.. 83. (a) Let A (Fig. 21) be a vertex of a triangle.
Extend the line segment A M beyond M such that
.

v,

.... ~4

A

~g. 21
:, A·'

,~e extensio.n has a magnitude I M N 1= ~ I AM I.
Tbe point 'N is the midpoint of the side opposite

"',vertex A, consequently, N must lie inside the
elreumscribed circle, that is, inside the circle of
t8-01657'

242


radius I OA I centred at O. Drop a perpendicular
OR from 0 on AN. There must be fulfilled the
inequality I AR I > I RN I. If LAMO:>- 90°,

then this inequality is fulfilled automatically. And
if LAMO < 90°, then I AM I - I MR I >

IMNI

+

IMRI

*

IAMI -

~ IAMI >

2 I MR I * I AM I > 4 I MR I. But R lies on
the circle a of diameter OM, hence A must be located outside the circle which is homothetic to the
circle ex with the ratio of similitude equal to 4 and
centre at M. Further, the point N must not get on
the circle a since otherwise the side of the triangle
whose midpoint it is, being perpendicular to ON,
would lie on the straight line AN, that is, all the
vertices of the triangle would be located on a
straight line. Consequently, A must not lie on the
circle which is homothetic to a with centre .of
similitude M and the ratio of 2. Thus, if we take
on the straight line OM points Land K such that
I LO I : I OM I : I M K I = 3 : 1 2, and construct
on LM as diameter the circle 1, on M K the circle 2,
then the required locus is represented by all the
points outside the circle 1 excluding the points
of the circle 2 except the point K (the point K
belongs to the locus).
(b) If 0 is the centre of the circumscribed circle, M the centre of mass of the triangle, then K
(see Item (a» is the intersection point of the altitudes of the triangle (see Problem 20 in Sec. 1).
But the distance from the centre of the circle
circumscribed about an obtuse triangle to the
point of intersection of the altitudes is greater
than the rad"us of the circumscribed circle. Consequently, the ~ertices of the obtuse triangle are
found inside th~ circle a, constructed on LK as
diameter, outside the circle 1 excluding the points
of the circle 2 (the vertices of obtuse angles lying
inside the circle 2).
84. Let ABC (Fig. 22) be the original regular
triangle, AlBIC. an arbitrary triangle with AIC I II

243


ACt AlB l U AB, 0 the centre of the circle, 0.
the intersection point of the altitudes of the triangle AIBtCt. Let LBOB t == q>. Since 0lB l II DB,
we have L. OB tOI = q>; since L. ClOlBI=L CtOB l =
120°, the quadrilateral C 10tOBl is inscribed
in a Circle, and, hence, L OlOC l = L 0tBlCl =
300 - cpo Thus L 0lOB = ~
1200
300 - cp =
1500 , that is, the straight line 001 is parallel to

+

+

A

Fig. 22
CB. To find the path which can be "covered" by the
point 0 1 , while moving along this straight line,
note that to determine the position of the point 0 1 ,
we draw through the variable point B l a straight
line parallel to DB to intersect the straight line
passing through 0 parallel to CB. Obviously, the

most remote points are obtained for the end points
of the diameter perpendicular to DB. Thus, M N
(the segment of the line parallel to CB, whose length
is 4R with the midpoint at 0) is a part of the locus,
the entire locus consisting of three such line
segments (with the end points of the segments
excluded).
85. If ABC (Fig. 23) is the given triangle, and
the vertex of the circumscribed rectangle AKLM
coincides with A (B on XL, C on LM), then L
belongs to the semicircle of diameter BC, the angles
ABL and ACL being obtuse, that is, L has two
16*

244


extreme positions: L 1 and L s, L L1CA = L LsBA =
90°, while the centre 0 describes an arc homothetlc
to the arc L 1L 2 with the centre of similitude at A
and ratio 172.

A

Fig. 23
A nsurer; if the triangle is acute, then the desired
set is a curvilinear triangle formed by the ares
of the semicircles constructed on the midlines as
diameters and faced inside the triangle formed by
the midlines; if the triangle is not acute, then the
required set consists of two arcs of the semicircles
constructed on two smaller midlines in the same
fashion.
86. If the first square is rotated about the point
M through an angle of 60° (see Fig. 24) either
clockwise or anticlockwise, then it must be entirely
inside the second square. Conversely, to each
square situated inside the larger square, and congruent to the smaller one, whose sides form angles
of 30° and 60° with the sides of the larger square,
there corresponds a point M possessing the needed
property. (This square is shown in the figure by
a dashed Iine.) This point is the centre of the
rotation through an angle of 60° carrying the
square ABeD into the square A1B1C1D 1; this


245

point can be obtained from 0 1 by rotating about 0
in the needed direction through an angle of 60°.
Consider the extreme positions of squares AtBtCtD t
(wpen two vertices are found on the sides or the

Fig. 24
larger square). Their centres serve as vertices of
the square KLRN whose side is equal to b -

(ya + 1)

i

aX

(the sides of the square KLRN are
parallel to the sides of the given squares, the centre
coinciding with the centre of the larger square).
The centres of another family of squares forming
angles of 30° and 60° with the sides of the larger
square also fill up the square KLRN. Thus, the
required locus consists of the union of two squares
one of which is obtained from the square KLRN
by rotating the latter about 0 through an angle of
60° in one direction, and the other by rotating
through an angle of 60° in the opposite direction.

+

1)
The problem has a solution if b:>- ; ('(a
(the points P and Q may be located on the boundary
of the squares).
87. There is only one such point, viz. the
centre of mass of the triangle (the median point).
It is easily seen, that in this case for any point N
on the boundary of the triangle we may take one
of the vertices of the triangle as a point P. Let us

246

take some other point M J • We assume that this
point is found either inside the triangle AMD or
on its boundary, where M is the centre of mass of
the triangle ABC, D the midpoint of A C. We~ draw
through M 1 a straight line parallel to BD ana take
the point of intersection of this line and A D as N,
denoting its intersection point with AM by M,."
Obviously, for any point P inside the triangle or
on its boundary the area of the triangle M1NP does
not exceed the area of one of the triangles AM IN,
MINe, M 2NB. It i8 also obvious that SAMIN <
SAMD=
1 ND 1

~

S.

=

»,

Further,

tb

if

SM.Ne _

en

SMDC -

a
~t
Finally,
~~ •
I ND I
(4 -.1:) Z
t
I AD I
al
<.
88. If A, B t C are the
ABC, then the angles of
2_z1

equal to

~, ~,90° + ~

I AD I = I DC 1=4,
I MIN I • I Ne ,

I MD I

I DC I

SM.NB _
SAMD

1 MIN I

IMDI

X

angles of the triangle
the triangle ABI are

(Fig. 25); consequently,

the sought-for locus is a pair of triangles two sides
of which are line segments, the third being an arc
which is a part of the segment constructed on AI
and containing an angle a/2.
89. We erect a perpendicular to BM at the
point M; let P denote the point of intersection of
this perpendicular and the perpendicular erected
to the original straight line at the point B. Let us
show that the magnitude I PB I is constant. Let
L MBC be q>; K and L denote the feet of the
perpendiculars from A and C on M B . By hy-

.

IMKI

pothesis, I KA I
J BC

I sin q>,

+ ILMII
LC

but

I LC 1=

I AK I = J BA J sin q>.

Hence,

r

=k,

J MK

I

:I·BA Isinq>

., BM I =+ I BL
I BC I sin q>

~C

I LM I

+ f BC I sin

iA~~i~W

247
k¢:!> IBMI±IBKI+

I BM I
sin q>

IHAI sin q>
( 1

TBAT

+

B~~~i~W)

I
I) ± ( I
I
=k<=>
I 11M I _ k I SA I · l BC I ¢:!> I PB I = k l BA I · i Be I
lin cp - I BA I + I BC I
IBA I + I BC I '

rig. 25
which was to be proved. Consequently, the soughtfor locus consists of two circles touching the
straight line AC at a point B and whose diameters
k I BA 1· BC I
are equal to I BA I + I Be l
90. Extend AQ beyond the point Q and take
on this ray a point M such that I QM I =

~ I AQ I

and a point Al such that I MAl I = I AM I; M is
t.he midpoint of the side Be of the triangle ABC;
LeBA I = LBCA, LABA I = 180 0 - LBAC.

248


Consequently, if we construct circles on AM,
MAt, and AA t as diameters, then the sought-for
locus consists of points situated outside the first
two circles and inside the third one.
91. Consider four cases: either the triangle ABC
is acute, or one of the angles A, B or C is obtuse.
In all the cases, it is possible to express the angles
of the triangle ABH in terms of the angles of the
triangle ABC.
92. If the end points of the rays do not coincide,
then the required locus is formed by the parts of
the following lines: the bisectors of the two angles
formed by the straight lines containing the given
rays, the midperpendicular to the line segment
joining the end.points of the rays, and two parabolas
(the parabola is a locus of points equidistant from
a given point and a given straight line). If the end
points coincide, then the desired locus consists of
both the bisector of the angle formed by the rays
and the part of the plane enclosed inside the angle
formed by the perpendiculars erected at the end
points of the rays.
93. Let A denote the vertex of the angle. It
is possible to prove that the centre of the circle
circumscribed about the triangle MON coincides
with the point of intersection of the angle bisector
A 0 and the circle clrcumscribedjibout AMN. Let a
be the size of the angle, r the rldius of the circle,
K the midpoint of AD. On the angle bisector AD,
we take points Land P such that I A L I =
r
, lAP I
r

+ . (1)

.

a)

. (%(1 sin 2"
(%(1 ·
sin 2"
sin 2 - SIn "2
The sought-for locus consists of the line segment KL (K not belonging and L belonging to
this set) and the ray lying on the angle bisector
with origin at P.
94. Let 0t' 0 1 denote the centres of the circles,
rt, rl their radii, M the midpoint of AB, 0 the midpoint of 0 10 8 • We have (~Y the formula for the


249

leJ)lth of a median, Problem 1t of Sec. t)
1
,;OlM 1 = ~ (2r~ + 21 0IB II - lAB I'),IO.MII=

t

+ 2 I O,A I' '~. (J 0 1°. 1 + 4 lOB II
'
2
(J OPt II + 4 I OA 1 (2ri

t

I AB (I),
-

2rl) ,

I 0IB II
IO.A

=

I' =

2
2rl)· Thus, 101M 1 -

J ~,M 1 =. rl - rl, that is (Problem t of Sec. 2)
pOInts M Iie on the perpendicular to DID". If the
tfreles have different radii and do not intersect,
'tlien the sought-for locus consists of two line
segments obtained in the following way: from the
line segment with end points at the midpoints of
the common external tangents, we remove the
points situated between the midpoints of the com.·on internal tangents (if M is a point on the line
segment with end points at the midpoints of the
common internal tangents, then the straight line
pwing through M perpendicular to OM does not
iJltersect the circle). In the remaining eases (the
~ireles intersect or are equal) the sought-fo.. locus
i. the, entire line segment with end points at the
midpoints of the common external tangents.
95'. (a) Since L FNB = 90°, LCNM = 135°,
i.FNM = 45° (we suppose that 'AM I> I MB I),
i. FNC = 90 0 and C, N, and B are collinear, and
80 forth.
(b) We consider the right isosceles triangle
ABK with hypotenuse AB (K lying on the other
aide of AB than the squares). The quadrilateral
ANBK is an inscribed one, LANK = LABK =
45°, that is, N K passes through M.
,
The desired locus is the midline of the triangle
ALB, where L is a point symmetric to the point K
wi,th respect to AB.
96. Let N denote the point of intersection of the
middle perpendicular and the tangent; 0 the centre
2
of the circle, R its radiua.. We have: I ON 1 2

250


+

I N A II = R2 I M N It - INA II = RI. Thus,
the required locus is a straight line perpendicular
to OA (Problem 1 of Sec. 2).
97. If 0 1 and 0t are the centres of the given
circles, QJ,., and Q, are the centres of the circles
circumscribed about the triangles ABCI and ABIC,
then 0IQIO,QI is a parallelogram. The straight
line QIQI passes through the midpoint of the line
segment 0 101 (the point D). The second point of
intersection of the circles circumscribed about
the triangles ABCI and ABIC is symmetric to the
point A with respect to the straight line Q1Qle The
sought-for locus is a circle of radius I AD I centred
at the point De
98. Let 0 1 and 0, denote the centres of the
given circles, rl and r, their radii. Consider two
right isosceles triangles 010 0 and 010,,0' with
hypotenuse 010,e The desireJ locus is two annuli
with centres, at 0 and 0' and the following radii:
external

~2 (rl + r t) and internal ~21 rl -

r't

I.

Let us prove this. Let M be a point on the circle
0t, N on the circle 0 .... If M is fixed, and N trav-

erses the second circle, then the vertices of the
right angles of the right isosceles triangles des-

cribe two circles of radius

~2 rtf which

are ob-

taioed from the circle 0, by rotating about M
through an angle of 45° (both clockwise and anticlockwise) followed by a homothetic transformation with centre of similitude at M and the ratio
V2/2. Let OM be the centre of one of those circles.
The point OM is obtained from 0 1 by rotating the
latter about M in the appropriate direction followed by a homothetic transfonnation with centre
of similitude at M and the ratio V2l2. But OM
can be obtained by corresponding. rotation and a
homothetic transformation with the centre of
similitude at 02. Consequently, when M describes

251


the circle 0lt OM describes a circle of radius V2 rl
2
with centre at 0 or 0'.
99. The union of the three constructed parallelo~ams represents the parallelogram circumscribed
about the given triangle separated into four
maIler parallelograms. It is easy to express the
relationships in which each of the diagonals under
consideration is divided by the other. diagonal in
terms of the segments of the sides of the larger
parallelogram.
If the parallelograms are rectangles, then, on
having translated two of the three considered diagonals, we get a triangle congruent to the given one,
and this means that the angles between them either
terual the corresponding angles of the triangle or
IUpplement them to 180°. The sought-for locus is
• circle passing through the midpoints of the sides
of the given triangle. °

100. We prove that :~~II = cos L.BAC, where
D is the point of intersection of AM with the circle.
~t 0 denote the centre of the circle, P the midJ'int of BC, K the midpoint of AN. The triangles
DOA and M KA are similar. Hence,

~~

t

=

I AKI

.
T"DOI = lOP II = I cos LBAC I· The desired
I OB
l()cus consists of two arcs belonging to two distinct

Circles.

£0, lOt. Let Bo. and Co be the midpoints of the
sides AC and AB, BB I and cel the altitudes, K
tbe midpoint of DE (Fig. 26), GK and CoN perpendicular to A B, B oM perpendicular to A C. Then

t.¥L I

iNM1 =

I GC I I
I KP I
I DC I
I COCI I = 1 CoC. I = I BC I (the last

ttquality follows from the similarity of the triangles
9CE and ABC, K, P and Co, C1 being the corre8pOnding points in those triangles). In similar

252


fashion, the middle perpendicular to D F intersects
.
I NL11
I BD I
MN at a point L 1 such that I N M I = I BC I'
that is, the points Land L 1 coincide.
A

8

D

Fig. 26
The sought-for locus is the straight line M N .
t02. It is obvious that any point of any of the
altitudes of the triangle ABC belongs to the required locus. We show that there are no other points.
Let us take a point M not lying on the altitudes of
the triangle ABC. Let the straight line BM intersect the altitudes dropped from the vertices A
and C at points M 1 and M 2' respectively. If the
conditions of the problem were fulfilled for all
the three points M 1 , M " and M, then the equalities LMAM1 = LMCM1t LMAM~ = LMCM 1
would hold, and then the five points A , M, M J' M t
and the potnt C1 symmetric to C with respect to
the straight line BM would lie on one circlet which
is impossible.
103. Note that if a straight line I possessing the
required property passes through M, then there
exists either a straight line 11 passing through M

253

ADswers, Hints, Solutions

wftij a vertex of the triangle or a straight line I,
_mng through M and perpendicular to a side 01
the triangle and possessing the same property.
tldeed, let the line 1 intersect the sides JIB and CB
of~he triangle ABC at points Co and A o' and let
there be a point B l symmetric to B with respect
,0 "-inside the triangle ABC. We rotate 1 about M
ab-tbat B 1 , moving in the arc of the corresponding
efrele, approaches AB or BC until the point Co
or B 0 coincides with the vertex C or A (and we get
~he line 1 ) or until !il gets on the corresponding
1
Side (and we get the Iine I.). Let a denote the set of
~h.. points of our triangle situated inside the
quadrilateral bounded by the angle bisectors
drawn to the smallest and largest sides of the
.M~Dgle and the perpendiculars erected at their
~points. (If the given triangle is isosceles, then
t.£ is empty. In all other cases a is a quadrilateral
or a pentagon.) The sought-for locus consists of all
~ .points of the triangle excluding the interior

J6lnts of

C1.

105. We have: I MB 12 = a l + cl cos' A =
.~ + c. - c' sin 2 A = 4 2
cl - at sin 2 C =
CI + 42 coa2 C = I NB 12 •
.: 107. Prove that the point symmetric to the
bltersection point of the altitudes of a triangle
with respect to a side of the triangle lies on the
Circumscribed circle.
109. Let H denote the intersection point of the
alitudes of the triangle ABC, AD the altitude,
g,; L, M, and N the projections of D on ACt CR,
RB, and BA, respectively. Take advantage of the
that K and L lie on the circle of diameter CD,
£ and M on the circle of diameter HD, and M and
N'on the circle of diameter DB.
t 1I , Prove that the radius of the circle circum~ibed about the triangle under consideration is
equal to the radius of the given circles, and these
9lkles are symmetric to the circumscribed one
~ith respect to the sides of the triangle.
it2. Let ABCD denote the given rectangle,

+

_t

254

Problems In Plane Geometry

and let the points K, L, M, and N lie on the
straight lines AB, BC, CD, and DA, respectively.
Let PI be the second point of intersection of the
straight line LN with the circle circumscribed
about the given rectangle (the first point is P).
Then BPI II KN, P1D II LM, and LBID = 90°.
Hence, KN 1. LM. In addition, LN..L KM;
thus, N is the intersection point of the altitudes
of the triangle KLM. Letjiow, for definiteness,
Land N be on the sides Be and DA. Denote:
I AB I = a, I BC I = b, I KP I = e, I PN I = y.
The straight line KN divides BD in the ratio

~: ~ ~; , counting from the vertex B. The straight
line LM divides BD in the same ratio.
113. The line segments I AP I, I BQ I, and
I CR I can be expressed in terms of sides of the
triangle, for instance: I A P I = b ~ t: •
114. Let M denote the midpoint of AD. Check
2
to see that I BF II
I FM 1 = I BM I'.
115. Draw through D a straight line perpendicular to the bisector of the angle A, then denote
the points of its intersection with AB and A C by
K and M, respectively, and prove that I A K I =

+

t

I AM I = b t:. Since I AC. I = I AB. I =
p - a, I ACt I = I BC I I = p (p the semiperim-

eter of the triangle ABC, and a, b, c its sides), the
points K and M are the midpoints of the line
segments CIC, and BIB I •
116. Prove that I forms with AD the same
angles as the straight line BC touching the circle.
Hence it follows that the other tangent to the
circle passing through D is parallel to I.
117. We construct the circle touching the
straight lines M N, A C and Be 80 that the points
of tangency P and Q with the lines A C and Be
lie outside the line segments eM and CN (this is
a circle escribed in the triangle M CN). If R is the


oint of tangency of MN with the circle, then
I MR I, 1NQ I = I NR I, consequently,
I MP I I NQ I; but we are given that
I M N I = I M A I I N B I. Thus, one of the
J'-Oints P or Q lies on the corresponding side, while
the other on its extension. We have: I CP I =

YMP II =
I MN =

+
+

I to I = ~ (I CP I + I CQ J) = ~ (I A C I +
I 'CB n, that is, the constructed circle is constant
for all the straight lines.
. t 18. If 0 is the centre of the circle circumscribed
about the triangle ABC, D the midpoint of CB, H
iIae point of intersection of the altitudes, L the
midpoint of AN, then I AL I = I OD I and, since
A·L is parallel to OD,OLbisectsAD,that is, Lis
IJDlmetric to 0 with respect to the midpoint of AD.
1t 9. Let BD denote the altitude of the triangle,
and I BD I = R Y 2, where R is the radius of
tle circumscribed circle, K and M are the feet
4JI~the perpendiculars dropped from D on A Band
JC, respectively, 0 is the centre of the circumf!I:.ihedcircle. If the angle C is acute, then L KBO =
goo - Le. Since BMDK is an inscribed quadrilateral, L. M KD = L DBM = 90° - L c. Hence,
i.MKB = 1800 - 900 - (900 - L C) = L C; con..
.-quently, BO is
perpendicular to
KM.
Bot S BKM =

~ I BD It

sin A sin B sin C =

R* sin A sin B sin C = ~ SABe- (We have used

tJle formula S = 2R2 sin A sin B sin C.) On the
MIler hand, if hi is the altitude of the triangle
f~:"·
1
1!KM drawn from the vertex B, then 2'" s =
t
1
T lAC 1·1 BD I = 8 B K M = 2 I KM I hi
t (hi SID
I
I
F I BD . B, hence, h1= 2 AC B = R; bearsin

256


ing in mind that BO J. KM, we get that the point

o lies on KM.

120. Note that the triangles ADK and ARK
similar since I AK 12 = 1 AC pa = IADI x
I A B I. If 0 is the centre of the circle circumscribed
about the triangle ABK, then LOAD
LADK =
90° - LAKB + LADK = 90 0 (LAKB was assumed to be acute; if L. A KB is obtuse, our reasoning is analogous).
121. Prove that the straight line parallel to BC
and passing through E divides the bisector of the
angle A in the same ratio as it is divided by the
bisector of the angle C.
122. If 0 is the vertex of the angle, A a point on
the angle bisector, B I and B I the intersection
points of one circle with the sides of the angle,
CI and C I (B 1 and Cion the same side) the intersection points of the other circle, then ~ABICI =

are

+

~AB.CI·

123. Take advantage of the fact that the common chord of the two circles passing through A, A1
and B, B l passes through the point D (Problem tlS
of Sec. 2).
125. If 0 is the centre of the circle circumscribed
about the triangle AMB, then L MAB = 90° LOMB = LBMC - 180°. The angle MAC has
the same size.
126. It is easy to prove that the circles under
consideration intersect at one point. Let us denote
this point by P. If the points are arranged as in
Fig. 27,
then L.PB.M = 180
LBB.P =
L PCtB =
1800 L PC1A
LPB 1A =
L. P A sA = 180° - L PA ,M, that is, the points
P, B I' M t and A. lie on one circle. In similar fashion, we prove that the points P, B" M t Cs lie on
one circle, consequently, the five points P, M, A I '
B I' C t lie on one and the same circle.
127. Prove that the sides of the triangle A IB IC1
are parallel to the corresponding sides of the triangle ABC.
128. Prove that as the straight line KL dis0

-

257

Answers, Hints. Solutions

places, the centre of the circle circumscribed about
KLB 1 describes a straight line.
129. Prove that any two line segments are
bisected by the point of their intersection.

......------...-aeC

w-=~-----

Fig. 27

130. If KN is a perpendicular from K on ARt
IKNI

IAKI

L. CAB=a, then IOMI = lAO ,=:;-0;

IAOI-IKOI

IAOI

IAOI-210MI siny

IAOI-2IAOI sin'T

IAOI

IAOI

cos a= : ~~ I' • Since the triangles ACB and
ACD are similar, it follows that KN is equal to
the radius of the circle inscribed in the triangle
A CD, and since K lies on the bisector of the angle
At K is the centre of the circle inscribed in the
triangle AC D. The proof for L is carried out in
a similar way.
13t. Denote by C1 and A 1 the midpoints of AB
and BC, by B' and A' the points of tangency of
the inscribed circle to A C and BC. Let, for definite17-01557

258


ness, c ~ b (c and b sides of the triangle A B C),
then the bisector of the angle A intersects the
extension of C1A t ata point K such that I AtK 1==
C - ; b , and the straight line B'A' must pass
through the same point K since the triangles KA IA '
and A 'B'C' are isosceles, I A 'C I == I B'C I,

I AIK I == I AlA' I, LA'AlK

= LA'CB'

132. Consider the angle at vertex A. Three
points B I , B 2' B 3 are taken on one side of the angle
and three points Cl' C 2' C 3 on the other side. From
Menelaus' theorem (Problem 45 of Sec. 2, Remark)
it follows that for the straight lines BIC t , B 2 C 2'
B aC3 to meet in the same point, it is necessary
and sufficient that the following equality is fulfilled:
AB2
GtC"
ABa
B"B t • C2A === B 3B l

•

C tC3
CaA

(1)

(the ratios are understood in the sense indicated in
Remark). Indeed, if the equality (1) is fulfilled,
then it follows from Menelaus' theorem that the
straight lines B 2C 2 and BaC s intersect the side
BtC I of the triangle ABIC I at one point.
133. Through A, draw a line parallel to BC
and denote by K and L the points of its intersection with AtG I and AtB lt respectively. We have:

IKAI
IACtl
ICBII
IAtGI
,
tBAtl = ICIBI ' IBIAI =rm·And,bYCevas
2
IACtl
I
44 ·

theorem
tBAll

(Prob em

In Sec.

),

lelRI

X

ICBtl
lAtGI • IBIAI = 1,
hence,
I KA I = I AL t.
But if A A 1 is the bisector of the angle KA tL, then,
since I KA I = I A L I, AA t is perfendicular to
KL, that is, AA 1 is the altitude 0 the triangle
ABC.
134. Let K be the point of intersection of .itA.
and BB lt H the intersection point of the altitudes

259


of the triangle ABC. The points A, K, H, and B
lie on a circle (the angles AKB and AHB are either
equal to each other or their sum yields 1800 according as the points K. and H are located either on
the same side of the straight line A B or on different sides). The radius of this circle is equal to the
radius R of the circle circumscribed about the
triangle ABC. If q> is the angle between AA 1 and
AN, then I KH I == 2R sin cpo
135. Let H denote the intersection point of
the altitudes of the triangle AIBIC t - The points
A It H, Bland C lie on the same circle, the points
Bit H, CI , and A also lie on a circle, the radii of
these circles being equal; the angles H B 1 C and
BBIA are either equal or supplement each other to
fSO°. Consequently, I HA I = I HC I. The converse
is false. For each point A I on the straight line
Be there exist, generally speaking, two triangles:
AIBIC I and AtB;C; (B 1 and B; lying on AC, CI
and C; on A B), for which the points of intersection
of the altitudes coincide with the centre of the
circle circumscribed about the triangle ABC, one
of them being similar to the triangle ABC, the
other not. For instance, if ABC is a regular triangle, At the midpoint of BC, then we may take
the midpoints of AC and AB as B t and C , and,
the points on the extensions of A C and A B teyond
C and B, as Bi and Cl , I CB; , = 1en I, I BC; , =
I BC I. The converse is true provided that the
points A l' B t , and Ct are situated on the sides of
the triangle ABC, but not on their extensions.
136. We prove that the' centre of the desired
circle coincides with the orthocentre (the intersection point of the altitudes). Let BD denote the
altitude, H the intersection point of the altitudes,
and K and L the midpoints of the constructed line
segments emanating from the vertex B, 1 BK 1 ==
I BL I = I, M the midpoint of BD. Then
I KH 12 =

12
t7*

+

I LH 12 =
1 M H 12
IBM 12
I MH 12 = II

+

1 KM 12

_

IBDI2

4

=

+

260

IBf'

r

Problems

in Plane Geometry

(I BH 1= II + I BH II - I BH J X
I BD I = l2 - I BH 1·1 HD I. It remains to prove

that the products of the segments of the altitudes
into which each of them is divided by the point of
their intersection are equal. We draw the altitude
AE. Since the triangles BHE and AHD are similar, we have: I BH 1·1 HD I = I AH 1·1 HE I,
137. We denote (Fig. 28): I BC I = a, I CA I =
b, I AB I == c. Through the centre of the inscribed

N

Fig. 28
circle, we draw straight lines parallel to A Band
BC to intersect A K and K C at points P and Q. In
the triangle OPQ we have: LPOQ == LANG,
I OQ I = p - c, I OP I == p - a, where p is the
semiperimeter of the triangle ABC. But, by hy-


261

pothesis, L.NBM = LABC, I NB I
p - a,
I MB 1== p ~ c. Consequently, ~POQ = ~NBM.
If we take on the straight line OP a point M I such
that I OM I I = I OQ I and on OQ a point N 1
such that I ON I I = lOP I, then ~ONIMI::Z
6NBM, and its corresponding sides turn out to
be parallel, i.e., BM " OM t and BN II ONI. Hence,
N 1M1 II N M. Let us prove that OK is perpendicular
to N1M 1. Since two opposite angles are right ones,
in tb.e quadrilateral OP KQ, the latter is an inscribed quadrilateral, consequently, L. 0 K P =
L.OQP. Further, L.KOP
L.OMtN I = L.KOP +
LOQP = L. KOP + LOKP == 90°,
and
this
means that OK .1 MINI.
t38. Let, for definiteness, P lie on the arc A C.
The points A, M, P, and N lie on one and the
same circle, hence, L N M P = L. NAP. Analogously, the points P, M, Q, and C are located on one
and the same circle, L PMQ = 180° - L.PCQ ==
t80° - LPAN = 1800 - LPMN.
t39. Let ABC be the given triangle (Fig. 29), H
the point of intersection of its altitudes. Note that

+

Fig. 29
the points symmetric to H with respect to its sides
lie on the circle circumscribed about the triangle
ABC (see Problem 107 in Sec. 2). If HI is a point
symmetric to H with respect to the side Be, then
the straight line 11 symmetric to I with respect to

262


the same side passes through H t . With 1 rotated
about H through an angle <p, the lineI, rotates about
HI through the same angle qJ in the opposite direction. Consequently, if P is the second intersection point of the line It with the circumscribed
circle, then the radius OP (0 the centre of the circumscribed circle) rotates about 0 through an
angle 2qJ in the appropriate direction. The same
reasoning holds true for the two other straight lines
symmetric to I. But if I coincides with an altitude
of the triangle, then the statement of the problem
is obvious (the point P coincides with the corresponding vertex of the triangle). Consequently, this
stat-ment is always true.
140. Let the points A, B, C, and M have the
following coordinates in the rectangular Cartesian
system: (ZI' YI)' (XI' Y2)' (xs Y3), (x, y), respectively, and let the coordinates of the point G are
t

(Xl +~+X3 • Y1 +~+ YS)

Then the validity of

the assertion follows from the identity 3 ( xXI+x.+x
3
3

)

2

= (x- XI)2+(x -

%2)2+ (x -

x3)2-

1

'3 «XI- X2)t+(X2- Xa)2+(xs-Xl)2» and analogous
relationship for the ordinates.
141. Ccnsider the case when the point M
(Fig. 30) lies inside the triangle ABC. Rotate the
triangle ABM about A through an angle of 600 to
bring B into C. We get the triangle AMI C which is
congruent to the triangle ABM; the triangle AMM 1
is equilateral, consequently, the sides of the triangle CM M t are equal to the line segments M A ,
MB, MC. The points M 2 and M 3 are obtained in a
similar way. The area of the hexagon AMtCM aB M I
is twice the area of the triangle A Be, that is,
equals a2 V 3l2. On the other" hand, the area of
this hexagon is expressed as the sum of the areas


263

of three equilateral triangles AMM l' eMM 3, and
BMM, and the three triangles congruent to the
desired one. Consequently, 3S + (I MA 12
I MB 12

+ I Me 12) va =
4

+

a2

V3.

Using the result

2
of Problem 140 of Sec. 2, we get 3S

+

(3d2

+

6

Fig. 30

as)

Y3= a2V 3, whenceS= ya (a 2 -

3d 2 ) . Other
4
4
12
cases of arrangement of the point M can be considered in a similar way.
142. Use the results of Problems 141 and 6 in
Sec. 2. Generally speaking, the sought-for locus
consists of a straight line and a circle.
143. Let (Fig. 31,a) 0 be the centre of the circumscribed, and 1 the centre of the inscribed cirele. From 0 and I, we drop perpendiculars ON,
OP, I L, and IQ on A Band BC. If a, b, c denote the
corresponding lengths of the sides BC, CA, and
AB, and p the semiperimeter of the triangle ABC,
then I B K I == 'c - b " 'B M I == I a - b I,
IBN' == c/2, I BP I = a/2, I BL , == I BQ I =

P -

b,

t
1 NL I = 2" I a

-

b I,

,PQ I

=

264

t

l

c -

Problems in Plano Geomelry
b

I (see Problem f8 in Sec. f). Consequent-

ly, if we draw, through 0, straight lines parallel
to the sides A B and Be to intersect the perpendir..
ulars dropped from I, then we get the triangle
ORS similar to BKM with the ratio of similitude
of 1/2. But the circle constructed on 01 as dia..
meter is circumscribed in the triangle DRS. Conseo
IJ

A ......- - -............. C

I

6

a

Fig. 3t
quently, the radius of the circle circumscribed
about the ~BKM is equal to 01. To prove the
second part of the problem, we note that if a line
segment OR! equal to OR is laid off on the straight
line OS, and a line segment OSI equal to OS-on
the line OR, then the line SIRI is parallel to KM
(Fig. 31,b); but LORIS!
LIOR t = LORS
LIOS = 90°, that is, SlRIl- 01.
144. Using the notation of the preceding problem, we draw through A a straight llne perpendicular to 01 and denote by D the point of its intersection wi th the straight line BC. Prove that
the difference between the radii of the circles circumscribed about the triangles ABD and A CD is
equal to the radius of the circle circumscribed
about the triangle BKM.
145. Let the sides of the triangle be equal to
a, b, and c, and b = (a
c)/2.

+

+

+


265

(a) From the equality pr = ~bhb (p the semiperimeter, r the radius of the inscribed circle, hb
the altitude drawn to the side b), we get:
f(a
kb
r=

+ b + c) == {-bh b ;

=

but a

+ c=

2b, hence,

3r.

(b) This assertion follows from the fact that

~ hb

and the median point divides each median

in the ratio 2 {.J
(e) Extend the angle bisector BD to Intersect
the circumscribed circle at a point M. If we prove
that 0, the centre of the inscribed circle, bisects
BM, then thereby our statement is proved. (We
draw the diameter BN, then the line joining the
centres of the inscribed and circumscribed circles is
parallel to NM, and LBMN = 900 . ) But
the triangle COM is isosceles since LCOM =
{

2'

+

(LC
LB). Hence, 1 CM 1 =
From the condition b = (a
c)/2,
by the property of an angle bisector, we get:
1 CD 1 = a/2. Let K be the midpoint of CB;
6.CKO = 6CDO (I cs I = 1CD I, LKCO ==
LOCD); hence it follows: LBKO = LCDM; in
addition, LDCM = LOBK == LB/2, I CD I =
I BK I that is, ~BKO == ~CDM, I CM I =
I 80 I, hence 1BO I = I OM I which was to be
proved.
(d) We take any point on the angle bisector.
Let the distances to the sides Be and BA be equal
LOCM

=

I OM).

+

to e, while to the side AC to y. We have: ~ X
by) = .. 8 4 => b (2% + y) = 28 4 =>
2z
y = hb •
(e) If L is the midpoint of BA, then the desired
quadrilateral is homothetie to the quadrilateral
BeMA with the ratio 1/2 (see Item (cj),

(az + ex

+

+

266


146. Let N denote the intersection point of
the common tangent with Be. It suffices to check
that I FN I
I NG I = I KN I
I NM I ==
I DN '·1 N E I. All the line segments are readily
computed, since I BD I = I CE I = p - b,

= Ib-

IDNI

r

p-a

c I , - - == r = - . - (ra the
tNEI
p
a
radius of the circle touching the side B.C and the
extensions of the sides AB and A C), and so on.
147. Through the vertices of the triangle ABC,
we draw straight lines parallel to the opposite sides
to form a triangle AlBIC! which is similar to the
triangle ABC. It is obtained from the triangle
ABC by a homothetic transformation with centre
at the centre of mass, common for the triangles
ABC and AtBtCl , the ratio of similitude being
equal to -2. The intersection point of the altitudes of the triangle ABC is the centre of the circle
circumscribed about the triangle A.Bl..Clo Consequently, the. points 0 (the centre 01 the circumscribed circle), G (the centre of mass), and H (the
intersection point of the altitudes of the triangle

I DE I

ABC) lie on a straight line, and lOG I

=~ I GH I,

G lying on the line segment OR.

148. In an acute triangle, Euler's line intersects the largest and the smallest ~ides. In an obtuse triangle-the largest and the middle sides.
150. Show that the required property is possessed by such a point P on Euler's line for which
I PO I = I OH I (0 the centre of the circumscribed
circle, H the intersection point of- the altitudes); in
this case, for each triangle the distance from the
centre of mass to the opposite vertex of the original triangle is equal to ~ R, where R is the radius
of the circle circumscribed about the triangle ABC,
and the straight line passing through the centre of
mass of this triangle and the opposite vertex of the
original triangle passes through the point O.

267

151. Let Ct denote the centre of the circle circumscribed about the triangle A P B, and C'}. the
point symmetric to C 1 with respect to AB. Similarly, for the triangles BPC and CPA we determine
the points A 1 and A 2' B t and B 2' respectively.
Since the triangles ACtB, AC 2 B , BAtC, BA 2C,
CBIA, CB 2 A are isosceles with vertex angles of
1200 , the triangles AlBICI and A 2 B 2 C 2 are regular
(see Problem 296 of Sec. 2). Computing the angles
of the quadrilaterals with vertices P, A~" B 2' and
C., we can prove that they lie on the same circle.
'Further, if H is the intersection point of the altitudes of the triangle APB, then, since I PH I =
'I GIC'}. I and, hence, PHC 2Ct is a parallelogram,
the straight line CtH (Euler's line of the triangJ~
APB) passes through the midpoint of PC'}.. But
PC I is a chord of the circle with centre at C I , consequently, CIH is perpendicular to PC 2 • Thus, the
three Euler's lines coincide with the midperpendiculars of the line segments PC t , PB 2 and PA 2 ,
'and since the points P, A 2' B 2' C 2 lie on the same
eircle, those lines intersect at its centre which is
the centre of the regular triangle A 2B2C2. It follows from the result of Problem 296 of Sec. 2 that
these three Euler's lines intersect at the median
point of the triangle ABC.
152. Let ABC be the given triangle whose sides
are a, b, and e (a > b > e), At, Bit Cl the points
of tangency of the inscribed circle, I the centre of
the inscribed circle, 0 the centre of the circumscribed circle. Since, with respect to the triangle
AIBIC l , I is the centre of the circumscribed circle,
it suffices to prove that the straight line 10 passes
through the intersection point of the al ti tudes of
the triangle AIBIC t . Layoff on the rays AC and
,BC line segments A K and BL (I A K 1= I BL I =
:t).., and O,D the rays AB and CB line segments AM
'and CN (I AM I = 1CN I) = b). As is known (see
Problem 143 in Sec. 2), the line 10 is perpendicular
to LK and MN, hence, LK" MN. Denote:
'- KLC = L BNM = <p. By the law of sines' for

268


the triangles KLC and BNM, we have:

I LC I

a-c

TKCT= b-c
I BN I
t BM I

a-b
b- c

=

=

sin (q>+C)
sin o
,

(1)

sin (B-q»
sin q>

(2)

Now, in the triangle AIBJ,C1 , we draw the altitude
to the side BIC I • Let Q be the point of its intersection with the straight line 10. We have to prove
that Q is the intersection point of the altitudes of
the triangle AIBICt • But the distance from I to
BICI is IIA I

I cos A l =

r sin: . Hence, the equal-

it.y I AIQ I = 2r sin: must. be true. The angles
of the triangle QIA t can be expressed in terms of
the angles of the triangle ABC and «p, namely,
o

180 - q>, L.QAtl =
. A
have to prove that 2 SID '2= .
L.QIA t =

LB-LC
2

· We

sin q>

B-C) <=>
sm( CP--2-

sin (cp + C) - sin (B - q» == sin «p. The
last
equali ty follows from (t) and (2).
153. When carrying out the proof, we make use
of the fact that if perpendiculars P K and PL are
dropped from a point P on the straight lines intersecting at a point M, then the points P, K, L,
and M lie on the same circle.
tM. Use the result of Problem 246, Sec. 1.
156. The distance between the projections of M
on AC and BC is equal to I CM I sin C. If K
and L are the projections of M on AB and BC,
then the projection of A B on the straight line K L
(this is just Simson's line) is equal to I AB I X
I cos L B K L I = I A B I I cos L. B M L I =, AB I sini-CBM = I eM I sin C.

'Answers, Hints, Solutions

269

157. Prove that the sides of the triangles
A1BCt , A 2B 2C 2 , and AsBaC a are correspondingly

parallel.
158. Prove that Simson's line corresponding to
Al is perpendicular to BIC I (the same for the other
points). Further it is possible to prove that Simson's line corresponding to the point Al passes
through the midpoint of AIH, where H is the
point of intersection of the altitudes of the triangle
ABC (see also the solution of Problem 166 of
Sec. 2). Consequently, Simson's lines are the alti;'udes of the triangle whose vertices are the midpoints of the line segments AIH, BIH, CIR.
,.Remark. We can prove that Simson's lines of arbitrary points AI, B, CI with respect to the triangle
ABC form a triang e similar to the triangle AIBIClt
ahe centre of the circle circumscribed about it coin~iding with the midpoint of the line segment joining
the points of intersection of the altitudes of the
f,riangles ABC and AIBICt •
1"59. First of all, we check the validity of the
following statement: if the perpendiculars drawn
to the sides (or their extension) of the triangle at the
pointe of intersection with a straight line meet at
• point M, then M lies on the circle circumscribed
about the triangle. (This statement is the converse
of the statement of Problem 153.) Consider the
parabola y = az 2 • An arbitrary tangent to it has
the form: y

=

ke -

k2
4a (the. tangent has only one

common point with the parabola, hence, the discriminant of the equation az 2 = ke
b is equal to
zero). This tangent intersects the x-axis at the
point z = k/4a. The perpendicular to the tangent
at this point is represented by the straight line
x
1 Consequently, all
1
1/= - - ( x- -k ) = - -

+

k

hell

4a

perpendiculars

k

+ 44
-.

pass

through

the point

270


(0; ~) (the focus of the parabola).

Now we use

the remark at the beginning of the solution.
160. Let ABC denote the given triangle, H
the point of intersection of its altitudes, At, BIt Ct
the midpoints of the line segments AH, BH, and
CH, respectively; AA 2 the altitude, A 3 the midpoint of BC. We assume, for convenience, that
ABC is an acute triangle. Since L BIAIC t =
LBAC and ~BtA2CI = 6B 1HC t , we have
LB1A 2 C1 = LBtHC I = 180
LBtAtC1 , that
is, the points At, B t , A 2' and C I lie on the same
circle. It is also easy to see that L B1A 3CI =
L BIHC I = 1800 - L BtAtC t , that is, the points
A lt B t , A s, and Ct also lie on one (that is, on the
same) circle. Hence it follows that all the nine
points, mentioned in the hypothesis, lie on one
and the same circle. The case of an obtuse triangle
ABC is considered in similar fashion. Note that the
nine-point circle is homothetic to the circumscribed
circle with centre of similitude at H and the ratio
of 1/2. (The triangles ABC and AtBtC t are arranged
just in such a manner.) On the other hand, the
nine-point circle is homothetic to the circumscribed
circle with centre of similitude at the median point
of the triangle ABC and the ratio of -1/2. (The
triangle ABC and the triangle with vertices at the
midpoints of its sides are arranged exactly in
such a way.)
161. Our statement follows from the fact that
D lies on the nine-point circle, and this circle is
homothetic to the circumscribed circle with centre
of similitude at H and the ratio of 1/2 (see Problem 160 of Sec. 2).
162. Our statement follows from the fact that E
lies on the nine-point circle, and this circle is homothetic to the circumscribed circle with centre of
similitude at M and the ratio of -1/2 (see Problem 160 of Sec. 2).
163. This distance is half the sum of the dis0

-

271


tances to BC from the intersection point H of the
altitudes and the centre of the circumscribed circle,
the latter being equal to half I HAl.
164. Let M 0 be the midpoint of HP, A 0 the
midpoint of H A, and the points A 0' A It and M 0
lie on the nine-point circle. Consequently, M also
lies on this circle since the hypothesis implies the
equality I MoB I·t HM 1= IAoH 1·1 HAl I, and
J[ is simultaneously either inside or outside each
of the line segments MoM and A oA1.
165. We prove that M and N lie on the corresponding midlines of the triangle ABC. If P is
the midpoint of AB, then LMPA == 2LABM =
LABC === LAPL. Let, for the sake of definiteness, ABC be an acute triangle, LC :> LA, then
LMNK == 1800 - LKNB == LKCB === LMLK
(we have taken advantage of the facts that the
points K, N, B, and C lie on the same circle and
that M L is parallel to BC). Hence, the points M,
L, N, and K lie on the same circle. Further

LLMK

LPMB

+

LNMK

= ~ LB

+

LBMK =fLB +LA. If o is the centre of the
circle circumscribed about the triangle LMK,
then LLOK === 2LLMK == LB + 2LA ==
1800 - LC
LA == 1800 - LLPK (LLPK ==
LAPK - LAPL == 180
2LA - LB ==
L C - LA), that is, 0 lies on the circle passing
through the points L, P, and K, and this is just
the nine-point circle.
166. Since the midpoint of FN lies on the
nine-point circle (see Problem 160 in Sec. 2), it
suffices to show that Simson's line corresponding to
:lhe point F also bisects FH. Let K be the projection
of F on a side of the triangle, D the foot of the
altitude drawn to the same side, HI the point of
intersection of this altitude and the circumscribed
circle, I HID I === I H D t (see the solution of
Problem 107 in Sec. 2), L the point of intersection

+

0

-

272


of Simson's line with the same altitude, and, finally, M the point on the straight line HH 1 for which
FM "KD; then ~FMHl = ~KDL (I FM I ==
, KD I), both of them being right-angled, and
L DLK = L. M HIF since the altitude of the triangle is the Simson line corresponding to the vertex
it emanates from, and we may use the statement of
Problem 154 of Sec. 2. It is also easy to show that
-+-.
the directions of HIM and DL coincide, that is,
FKH L is a parallelogram whence there follows our
statement.
167. In Fig. 32: 0 is the centre of the circumscribed circle, A h B I , C l the midpoints of the sides,
A

_ _ _----1--..-----..

c

8

Fig. 32
Land K are respective projections of A and B on
I, M the point of intersection of the straight lines
passing through the points Land K perpendicular
to Be and CA. For definiteness, the triangle ABC
is acute-angled. First, we prove that Cl is the
centre of the circle circumscribed about the triangle KLM. The points A h 0, K, CI , and B lie
on the same circle. Consequently, L CtKL =
L. OAtC l = 90° - L C; in similar fashion,
L C1LK = 900 - L C. Hence, I KC l I = I CIL I,

273


LLCtK == 2LC, and since LKML = LC, our
statement has been proved. Further, KM is perpendicular to AIC h I KC I I = I C1M I, hence,
LCIMA I = LCIKA I == 1800 - LB, that is, M
lies on the circle circumscribed about AIBIC t •
168. Let H denote the intersection point of the
altitudes of the triangle ABC, and As, B 2 , C,
the midpoints of the line segments AR, BR, CH,
respectively. Note that the triangles ABICI ,
AIBCl , AlBIC are similar (the corresponding vertices being denoted by the same letters), A 2' B 2'
and C 2 denoting the corresponding centres of the
circles circumscribed about them. First, we prove
the following assertion: three straight lines passing
through the points A 2' B 2, and C2 and occupying the
same positions relative to the triangles AB1Clt
AIBC I, AlBIC meet in a point on the nine-point
circle. Note that the straight lines A sB 1 , B 2B,
and C~B I are equally arranged with respect to the
triangles ABle AIBC It and AlBIC and intersect
at the point B I ying on the nine-point circle. Since
the points A 2' B 2' C 2 lie on the nine-point circle,
it is obvious that the three lines obtained from the
straight lines A 2B l , B 2B, and C B l by rotating
them about the points A 2' B 2t anJ C2' respectively, through the same angle, also intersect at one
point located on the nine-point circle. Let now P
be the intersection point of the Euler lines of the
triangles ABICl, AJ,.BCl.AIBIC. Denote: LPA1A =
cpo For the sake ot convenience, we assume that
ABC is an acute triangle, and the point P lies on
the arc BIA 2 of the nine-point circle (see Fig. 33).
Then LPA 2 A t = 1800 - CPt LPAtB l = 1800 cp - LB IA 2A I = 1800 - cp - LBICIA I
2LC - cp, LPA 2 Cl = 1800 cp
1800 0
2LB == 360 - cp - 2LB. Since the chords PAl'
PBI' and PC I are proportional to the sines of the
angles subtended by them, it remains to prove that
one of the three quantities: sin cp, sin (2C - q»,
- sin (2B
cp), (in our case the first one) is equal
to the sum of the two others, that is, sin cp =

l

,

+

+

18-01557

274


+

sin (2C - <p) - sin (2B
<p). But in the triangle
AA aH1 : I AA a 1= R, I AH I I = 2B cos A (R the
radius of the circumscribed circle, R cos A the
distance from the centre of the circumscribed circle
At to BtCl ) , LHIAA a = LA
2LB - 180°.

+

A

Fig. 33
By the law of sines for the ~AA 2"1' we have:
2 cos A
1
.
- . CP
.
SID
sin (2B+A+ cp )=>-SIO (2B+2A+cp)sin (2B
q» == sin q> => sin (2C - cp) sin (2B
q» = sin cp, which was required to be
proved. Thus, we have proved the statement for an
acute triangle. The case of an obtuse triangle ABC
can be considered exactly in the same way.
169. Let ABC be the given triangle, At, B l ,
and Ct the midpoints of the corresponding sides.
Prove that the circle passing, for instance, through
the vertex A and satisfying the conditions of
the problem pasaes through the points of intersection of the bisectors of the internal and external
angle A and the midline BtC I - Hence, for all
points M of this circle the equality I BIAI I:
I CIA! I = IB1A I : I CIA I = b : a is fulfilled (sec

+
+


275

Problem 9 in Sec. 2). Thus, if M 1 and M 2 are intersection points of two such circles, then
I A 1Ml I : 1 R 1MI I : ICIM I 1 = a : b : c (the same
for the point M 2), therefore M I and M I belong to a
third circle. In addition, M land M I belong to a
straight line for all points Mof which the equality

+

+

2
(c' b2 ) I AIM F~
(a2 e2 ) I BIM 1
(b2 - a2 ) I CIM 12 = 0 is fulfilled (see Problem 14
in Sec. 2 and its solution). This line passes through

the centre of the circle circumscribed about the
triangle AIBIC I and through the point of intersection of its medians (check this, expressing the
lengths of the medians in terms of the lengths of
the sides), that is, it coincides with the Euler line
of the triangle AIBIC I, and, hence, with that of
the triangle ABC.
170. (a) As it was done in the preceding problem, we can prove that these three circles intersect at two points M I and M 9' and I AMI I
I BM I I : I CM I I = be : ac ab (the same for the
point M 2).
(b) Follows from (a) and Problem 14 of Sec. 2.
(c) Prove that if M is inside the triangle ABC,
then LAMIC = 60° + LB, LBM1A = 600
Le, LCMIB = 60° + LB (for this purpose, use
Bretschneider's theorem-Problem 236 of Sec. 2).
171. Take on BC a point Al and on BA a
point CI such that I BA l I = I BA I, I BCI I =
I Be I (the triangle AIBC I is symmetric to the
triangle ABC with respect to the bisector of the
angle B). Obviously, BK bisects AICJ . We construct
.~wo parallelograms BAIMC I and BCND (the
corresponding sides of the parallelograms are parallel, the points B, K, M, and N are collinear);

+

I cs I
IAKI
IKCI

IBel
IBCI2
I AA 1 I IBAII = IBAI ,consequently,
IABI
IABI 2

= ICNI

=

tBCra

172. We have (Fig. 34) LFEIA = LEDF
LA, hence, I AF I = I ElF I, LFE1N =

=

276
LFDB = Le,

LEIFN

=

LA.

~EIFN is similar to ~ABCt
tACI
IABI ' LAFN = tBO° -

Consequently,

IAFI
IFNI =

IEIFI
IFNI

==

LA. Now, we can show

A

c
Fig. 34
that A N is symedian. To this end, consider the
parallelogram ACAlB; AA 1 bisects BC, the triangle A CA I is similar to the triangle A FN, hence
LNAF = LAlAC.
t 73. The Apollonius circle passing through the
vertex B of the triangle ABC is the locus of points
.
M for which

IAMI

IABI

IMCI == IBCt (Problem 170, of

Sec. 2, Solution). Consequently, if D is the point of
intersection of this Apollonius circle and the circle
circumscribed about the triangle ABC, then the
straight line BD divides A C in the ratio SSBAD:=
BCD

tABt·IADI _ IABI4
ICBI·ICDI - ICBra'

277


t 74. Let N denote the point of intersection of
BQ and CD, 0 the centre of the circle, R its ra-

dius. Note that LNBe = -}LPMQ. (If Q lies on
the line segment NB, then LNBC = 90°
f
1
LQBP == 90o 2 L QOP ="2PMQ.) Hence, the
triangles NBC and POM are similar, I CN I ==
R
IPDI
IBPI
1
I BC I IPMI = R 1PM 1 == R 1AB 1 == 2 1BP 1 ==

t

"2 I CD I·
t 75. Let H be the intersection point of the altitudes, 0 the centre of the circumscribed circle, B 1
the midpoint of CA. The straight line M N passes
through K, which is the midpoint of BH, I BXI ==
I BIO I. Prove that the line M N is parallel to DB
(if LC> LA, thenLMKN == 2LMBN = LeLA = LOBH).
t 76. Let the straight line A M intersect for the
second time the circle passing through B, C, and
M at a point D. Then LMDB = LMBA==
LMAC, LMDC == LMBC = LMAB. Consequently, ABCD is a parallelogram.
177. From the solution of Problem 234 of

.

ILMI

ILNI

Sec. 2 It follows that IMKI = INKI. We may
assume that l passes through N. Appyling the law
of sines to the triangle N K P and replacing the
ratio of sines by the ratio of the corresponding

INKI sinLNKP
I NP I =
sinLKPN ==
INKI sinLNKM
INKI
sinLKMA
= IKMI I NMI and so forth.

chords, we have:

t 78. Let 0 denote the centre of the inscribed
circle, K and L the points of tangency with the
sides A C and AB. The straight line passing through
N parallel to BC intersects the sides AB and A C
at points Rand M. The quadrilateral OKMN is


278

an inscribed one (LONM = LOKM = 90°);
consequently, LOMN = LOKN, analogously,
LORN = LOLN, but LOLN = LOKN, hence
LORN = LOMN, and the triangle ORM is isosceles, ON is its altitude; thus I RN I = I N M I.
179. If I BC I = a., I CA I = b, I AB I = c,
then, as is known (see Problem 18 in Sec. 1),
a+b-c
I MC I 2
· We draw through K a straight
1ine parallel to A C, and denote its intersection
points with AB and BC by Al and Cit respectively.
The circle inscribed in the triangle ABC is an
escribed one for the triangle AIBCI (it touches
AIC I and the extensions of BA I and BC I ) . But the
triangle AIBCI is similar to the triangle ABC.
Consequently, the circle escribed in ABC will
touch A C at a point N; let Rand L denote the
points of tangency of the circle with the extensions
of BA and Be, respectively. We have: I BR I =
1
I BL I = 2(a + b c), hence I AN I = I AR I =

+

I RB I - I BA 1=

a+:-c =

I Me I.

180. Draw through K a straight line parallel
to BC. Let L and Q denote the points of intersection of the tangent at P with the line BC and the
line constructed parallel to it, and N the point of
intersection of A K and BC. Since I CN I =
I BM I (see Problem 179 of Sec. 2), it suffices to
prove I N L I = I LM I; but I P L I = I LM I,
hence, have to prove that I PL I = I N L I. Since
the triangle PLN is similar to the triangle PQK ,
in which I PQ I = I QK I we have I P L 1 =
I NL I and I CL I = I LB I·
t 81. Let M and N denote the points of intersection of the straight line L K and the straight
2
lines l and CD. Then I AM 1 = I M L 1·1 M K I.
From the similarity of the triangles KMB and
DKN

.

It follows that I MK 1=

IKNI·IMBI
IDN,·


279

Since the triangles CNL and MLB are similar, we
.
_ ILNI·IMBI
have. I ML I ICNI
·
IKNI·ILNI
Thus, I MK I J ML 1 =
ICNI.IDNI X
1MB 12 = 1MB 12 , that is I MA 12 = 1MB 12 ,

I MA I = I MB I.

182. Let B be a second common point of the
circles, C the point on the straight line A B from
which the tangents are drawn, and, finally, K the
point of intersection of the straight lines M Nand

Fig. 35
PQ (Fig. 35). Making use of the law of sines and
the result of Problem 234 in Sec. t, we get:
I PM I
I PM I
sin L PBM
IbM I
I MAl
sin L PBM
I MAl :: sin L BPM X
sin L PBM .. /TCBT sin L PBM
I MA I
V ICAT' sin L BPM' Thus, denoting the angle AMB by ex and the angle APB


280
by

P

(ex and

P

constant) we get:

.. ;-TCBT · sinsin P ·
(a,+P)
V TCAT
I AN I
I NQ I

.V

Analogously ,

I PM I -IMAIwe

rl"CAT sin ~
TCBT· sin (a,+P) • But, by

find:

Mene-

laus' theorem (see Problem 45 in Sec. 2), : ~ : X
fAN I
lQK I
lNQI IKPI
1. Hence, IQKI/1KPI=1.
183. Through the point M, we draw a straight
line parallel to A C to intersect the straight lines
BA and BC at points Al and Ct. We have:

LA1KM = 90 0 - LDKM = 90° - LKBD =
LBAD = LKA 1M; consequently, KMA I is an
isosceles triangle, and I A 1M I = I M K I. Analogously, I MC I 1= 1ML I; but I KM I = I ML I,
hence I AIM I = I MC I I, that is, the straight line
BM bisects AC.
184. Let M denote the point of intersection of
N D and A B, and P the point of intersection of the
tangents to the circle at the points A and D.
Since the straight lines Ne, AB, and PD are
parallel, from the similarity of the corresponding
triangles we get:

IANI

IAMI=IDPI·,NPI'

(1)

I MB I ' M D I
I NC I - I ND I

(2)

I AP I
I ND I '

I API
IMBI=INC, INPI ;
but I DP I = lAP I, I sc I = I AN I. Consequently, the right-hand sides of the expressions (1)
and (2) are equal to each other, that is, I AM I =

IMBI.

Answers, Hiots, Solutions

281

185. We assume that D is the midpoint of CH,
and A D intersects the circle for the second timo at
a point K. Let us prove that the tangents to the
circle at the points Band C intersect on the straight
line M K.
Consider the quadrilateral CMBK. For intersection point of the tangents to the circle at the
points C and B to lie on the diagonal M K, it is
necessary and sufficient (see Problem 234 of See. 1)
ICMI _ IMBI .
ICMI _ IABI_
that ICKI - IBKI'
but ICKI ICXIIBDI
ICDI
IACI
IMBI
IDKI = IDKI = 18KI = IBKI· (In the first
and last equalities we have used the fact that
I CM I = I AB I, I AC I = I MB I since AM is
parallel to CB, in the second and fourth equalities-that the triangle A BD is similar to the triangle CDK, and the triangle ADC to the triangle
KDB, in the third, the fact that AD is a median.)
186. Let 0 denote the centre of the circle, Nt,
Mit Ph R 1 the points symmetric to the points N,
M, P, R with respect to the straight line OA t respectively,' K the point of intersection of the straight
lines NIR~· and QS. We have to prove that the
points R t , S, and K coincide. The points N" M I ,
and B lie on the same straight line symmetric to
tho straight line NMC; the points N t t PI' R 1
also lie on a straight line symmetric to the straight
line NPR (Fig. 36). The points B, s ; Q, and K
lie on one circle since LBN)K = LM 1NI P l =
LMNP = LPQM = LBQK. The points B, N t t
Q, and R 1 arc also on one circle since LN 1 R 1B =
LN1P1P = LN 1QP = LN1QB. Consequently,
the five points B, Nt, 0, R lt and K are located on
the same circle; but the points N t t R., and K are
colli ncar, hence R I and K coincide.
t87. Let us confine ourselves to the case when
A BC is an acute triangle. Consider the parallelogram AlMON (M and N on AtB) and AlGI' respectively). Since AID forms with AIC , and AlB.


282

angles of (900

LB) and (900
AIM I
cos B
-

I AIM I
I
I A1N liMO I

-

LC), we have

I AIL I
cos C = I A1K I •

I
N

--+-I
t
I

o
Fig. 36
188. The statements of the problem follow
from the fact: if a circle is constructed on each side
of the triangle so that the sum of the angular values
of their arcs (located on the same side with the
triangle) is equal to 2:t, then these circles have a
common point.
189. Take the points E 1 and F I symmetric to
the points E and F with respect to AB. Then the
problem is reduced to a particular case of Problem
186, Sec. 2.
t90. On the extension of AC beyond the point
C, we take a point M such that I CM I = I CB I;
then E is the centre of the circle circumscribed
about the triangle AMB (I AE I = I BE I,
LAEB == LACB = 2LAMB). Hence it follows
that F is the midpoint of AM, and D F bisects the
perimeter of the triangle ABC. In addition, D F
is parallel to B M, and B M is parallel to the bisector
of the angle C of the triangle ABC that is, DF


283

is the bisector of the angle D of the triangle D K L,
where K and L are the midpoints of AC and CB,
respectively.
t 91. Let the straight line intersect the sides A C
and AB of the triangle ABC at points M and N.
Denote: I AM I
I AN I = 21. The radius of the
circle with centre on MN touching AC and AB
is equal to SAM;ri/l, and, by hypothesis, SAMN/l =
S A BC/P = r, where p is the semiperimeter and r
tlie radius of the circle inscribed in the triangle

+

ABC.
t 92. Prove that in the bomothetic transformation with centre at M and the ratio of similitude of
-1/2 the point N goes into I (obviously, this homothetic transformation carries the point I into
S). Let ABC be the given triangle, A o, BQ.' and
Co the midpoints of the sides BC, CA, and. AB,
respectively, At a point on the side BC such that
AA 1 divides the perimeter into two equal parts.

It is easy to see that At is the point of tangency
with the side BC of the escribed circle which also
touches the extensions of the sides A B and A C, A I
the point of tangency of the inscribed circle wi th
the side BC. We have: I BA I I = I CA J I. We
erect at point A 2 a perpendicular to BC and denote
by D the point of its intersection with AA t • Repeating the reasoning for the solution of Problem 179
of Sec. 2, we prove that I Aal I = lID I. Consequently, the straight line A 01 is parallel to A A 1- If
we carry out the homothetic transformation mentioned at the beginning, then the straight line A Al
goes into the line A 01. In similar fashion, two other
straight lines bisecting the perimeter go into B 01
and Col, respectively. Hence, all these three lines
intersect at such a point N which goes into I in
this transformation. This implies the statement of
the problem.
.
193. (8) USIng the formulas

S
r=-,;,

abc

R= 4S '

S=Vp(p-a)(p-b)(p-c), where S is the area


284

of the triangle ABC, we easily prove the given
relationship.
(h) Use Leibniz's formula (Problem 140 in
Sec. 2), taking the centre of the circumscribed circle as M.
(c) Use Leibniz's formula (Problem 140 of
Sec. 2), taking the centre of the inscribed circle as
M. To compute, for instance, I MA 12 , we drop a
perpendicular M K on AB; we have: I M K I = r
I AX I = 1J - a; hence, I AM II = (p - 4)2 +,
r2 , 1 MB II and I MC 12 are computed in a similar
way. For simplifying the right-hand side, use the
result of Item (a).
(d) Let M denote the intersection point of the
bisector of the angle B and the circumscribed circle.
If 110 I = d, then I·BI , , 1M I = R2 - tiI'. The
triangle IeM is isosceles (11M I = 1eM I) since
1
1
LC1M= '2 (LB
LC) and LICM = 2(LB

+

+

LC). Consequently,

I BI 1·1

RS - d'

= I BI I· 11M 1=

cu I=_r_.2Rsin!!..=2Rr.
. B
2
sln

2

(e) Can be proved in much the same way 8S
Item (d).
(f) The distance between the projections of I
and fa on AC is a. We take a point K such that,
IK II AC, 14K 1. AC. In the right triangle IKI a ,
1
we have: LKIla = 2LA, 11K I = at IlaK 1=
11K P'
a
ra-r. Thus, 1 11 a12 =
A
-=--A 2 I I K Ix
cos2 sin
2
A

tanT=4R(ra-r).
194. Through the point 0, we draw straight
lines parallel to AB and AC and denote by Land
K the intersection points of these lines with the


285

perpendiculars dropped from. I on AB and AC,
respectively. Let us prove that the triangles ABIC I
and OLK are similar. We have: L BIAC I == L LOK,
(l

be
be
c
I AB11==-+ I ACII
e a
c a ,IOL l=P--2=
t
b
1
2:(a+b),
10K I =P-T="2 (a+c);
thus,

=-+

I ABI I
lOLl

=

I AC 1 I
lOKI

2bc

.

But oi, IS
the diameter of the circle circumscribed about
the triangle OLK. Consequently, ,BIC 1 I =
==(c+a)(b+a)·

2bc

(c

2bc

+ a) (b + a) I LK 1 = (e + a) (b + a)

abc
(c+a) (b+a) R· lOla

.

I 01 a I sin A==:

I·

196. Prove that the area Qa of the triangle with
vertices at the points of tangency of the escribed
circle centred at I a can be computed by the formula
r
S~BC , were t h e notation
h
.
Qa= SABC..!!... =
2R
2R(p-a)
is the same as in Problem 193 of Sec. 2. Analogous
formulas can be obtained for the areas of other
triangles. (See the solution of Problem 240 of
Sec. 1.)
197. Let 0 be the centre of the circle circumscribed about the triangle ABC, B the midpoint
of A C, N the point of tangency 0 the inscribed
circle with AC. Then I AN I = p - a, I eN I ===
p - c (see Problem 18 in Sec. 1), I ON 12 =
IOB I 12
I BIN 12 === 1 AO II I AB I 12

1

+

I BiN

,3 =

R3 -

~+

(p -

a -

~)3

+

R3

(p - a) (p - c). We then determine the squares
of the distances to the other points of tangency and
add them together to get the desired sum; it is
equal to 3R2 - (p - a) (P.- c) - (p - c) X
(p - b) - (p - b) (p - a) = 3R2 - M. Making
use of Hero's formula for the area of a triangle
and the fonnulas S = pr and S = abcl4R, we get:


286
~ =

(p - a) (p - b) (p - c)/p, 4Rr= abclp . AddIng together the last equalities and using the identity (p - a) (p - b) (p - c)
abc == p «p- a) X
(p - b)
(p - b) (p - c)
(p - c) (p - a)) ==

+

+

+
+

pM, we find M == 4Rr
Answer: 3R2 - 4Rr - r2 •
198. The product of the lengths of the line segments from the vertex A of the triangle ABC to the
points of intersection of the side AB with the given
circle is equal to the product for the side A C.
Each:of these line segments can be readily expressed
in terms of the sides of the triangle and the
chords under consideration. Thus, we obtain a
system of three equations enabling us to express
the chords in terms of the sides of the triangle. To
avoid the looking over of variants, it is convenient to choose a certain direction of traversing the
triangle and regard the line segments to be directed
and their lengths to be arbitrary real numbers.
199. Let K 1 and £1 be points on Be; and BAt
respectively, such that K 1K 1/ L 1L 1/ BIB. It suffices
to prove that the triangles BK1K and BLIL are
·"1
th t "
I BK 1 I
I ELI I
h
SImI ar,
a IS, I KIK I
I LIL I· Weave:
I BK 1 I
I BIK I
I K1K I
I A1K I
, BA I I = I BIA I I ' I BB I I = I BIA I I ' and
by the property of an angle bisector (Problem 9 in
I B 1K I I BA I I
I CE I t
Se 1) I BK I I
c.
, I K 1K I - I A 1K I I BB I I
I CAl I X
I BA I I
c I CB t I
ca
I BB I I
b I BB I I - (c+a) I BB I I · The last
expression is symmetric with respect to a and c,
'BL I
d
an, h ence, I. t·IS a Iso equa I t 0 I LIL II .
r2 •

200. Let LKAL = LKLA == cp, LKCL =
LLKC == '1'. Then LBKL = 2q>, LBLK = 2""
2cp + 2'l' == 1800 - LB. If Q is the point of intersection of AL and KG, then LAQC = 1800 1
(cp +,p) == 90 0 +2LB. Through M, we draw a


287

straight line parallel to BC to intersect K C at a
point N, then M Q is the bisector of the angle AMN
and LAQN = 90° + ~ LB. Hence it follows that
Q is the intersection point of the angle bisectors of
the triangle AMN (see Problem 46 in Sec. t);
hence the triangle AMN is similar to the triangle
KBL, and the triangle KMN is similar to the
triangle KBC. Let I AK I = I KL I = I LC I == x,
I AM I
y, I MN I
e, Then _z_ = _Y_,
a-x

c-x

=..:.. whence y == a.
c-x
a'
y-x

201. Let B 1 be the midpoint of AC. Extend the
angle bisector to intersect the perpendicular, erected at the point B l to AC, at a point B t . The
point B 2 lies on the circumscribed circle. Through
the point M, we draw a perpendicular to A C;
let L be the point of its intersection with AC, K
that with BB t , then I KM I = I ML I. We draw
through the point K a straight line parallel to A C
to intersect the straight lines AB and BC at points
D and E, respectively. If G and F are the projections
of D and E, respectively, on AC, then M is the
centre of the rectangle GDEF, the triangle DME
being similar to the triangle A B 2C (the triangle
DME is obtained from the triangle AB 2C by means
of a homothetic transformation with centre at B).
.
_ ILCI
ILFI
We have.
cot LMCL IMLI == IMLI
IFCI
IABll
IFCI
B
IMLI = IB B
21
2 cot C.
EF1= cot "2
I 21
If now B' is the foot of the angle bisector, P and
T are, respectively, the projection of Nand B'

+

+

+

IPC,

IPTI

+

on BC, then cot LNCB = INPI = INPI
ITCI
IBPI
ITCI
B
INPI = INPI
2 iB'Tl == cot 2"
2 cot C,
that is, LMCA == LNCB.

+

+

288


202. (a) This well-known problem has many
proofs. Consider one of them based on the following test for the congruence of triangles. Two triangles are congruent by one equal side, an equal
angle opposite to this side, and an equal bisector
of this angle. Let us prove this test. Consider two
triangles ACB and ACB I in which LB = LB l
(B and B I lying on the same side of AC). These
triangles have a common circumscribed circle. We
may assume that Band B I lie on the same side of
the diameter of this circle which is perpendicular to
A e. Let the bisector of the angle B intersect A C at
a point D, and the bisector of the angle B J. at a
point D t , M the midpoint of AC, N the midpoint
of the arc A C not containing the points Band B I •
The points B, D, and N are collinear. as well as
Bit D t • and N. Let Band B J be non-coincident,
and, hence, D and D 1 are also non-coincident.
Suppose that I MD I> I MD I I; then I BN 1 I DIN I. Consequently,

I BID I I = I BIN I - I ND I > I BN I I ND I = I BD r which is a contradiction. Let now

the bisector AA I in the triangle ABC be equal to
the bisector eel. Apply the test just proved to the
triangles BAAl and BeC I •
(b) If both bisectors of the external angles A
and C of the triangle ABC are found inside the
angle B, then the proof can be carried out just in
the same manner as in Item (a).
Let these bisectors be situated outside the angle
B. We shall assume that I BC I > I BA I. Take on
CB a point B 1 such that I CB I I = I AB I I. Let
LB1AC = LBCA = a. t LBlAB = cp, L the intersection point of the bisector of the external angle
C and A B, M the intersection point of the bisector
of the external angle A and CB. The rest of the
notations are clear from Fig. 37. By hypothesis,
I CL I = I AM I, in addition, I eLf I = I AMI I,
since BtAC is an isosceles triang e, I CM; I =
I AM I since the triangles CL1Mi and AM1M are
congruent. Further, I cu; I > I CM; I, since

Answers, lliols, Solutions

289

LM;M;C> LM;CA > 900 • On the other hand,
the points C, A, L, and Mi lie on the same circle
in which the acute angle subtended by LC (LLA C)
is greater than the acute angle subtended by M;C.
Hence, I AM I = I GM; I < I GM~ I < I CL I.
But this is a contradiction.
In the general case, the equality of the bisector S
of the external angles does not imply that the

Fig. 37
triangle is isosceles. Problem 256 of Sec. 1 gives
an example of such a triangle.
203. Let ABC be the given triangle, AA I, BB I ,
CCI the angle bisectors. If I AIB I I = I AlGI I,
then either LAIBle = LAlCtB (in this case the
~ABC is isosceles) or LAIBle
LA1CIB=
1800. In the second case, we rotate the triangle
AlBIC about the point At through an angle BIAICt •
As a result, the triangles AICIB and AlBIC tum
out to be applied to each other and form a triangle
similar to the triangle ABC. If the sldes of the
triangle ABC are a, b, and c, then the sides of the

+

t 9-01561

290.

.

ac

ab

obtained trtangle are equal to b+c' b+c and

+

ac
ab
.
.
.
.
a+b
a+c· B earmg In min d t hat th e triang1es
are similar, we get:
c
b
a
a+b
a+c = b+c
<=> b3 + eS - as + b2 e b2a+ eZb+ e'a- a2b- ate
+abe=O.
(1)

+

+

Let us denote cos L BAG = x, By the law of cosines,
b2+c2_-al=2bex. Multiplying the last equality,
in succession, by a, b, and e and subtracting it
from (1), we get:
2 (b+e) x
2x(a+b+c)+a=O<=> a= 2x+1 •
Since 0 < a < b+c, we have
1

-4'<%<0.

(2)

Expressing a in the law of cosines in terms of
% and denotinz blc = A, we obtain for A
1
the equation (4x + 1) A - 2A (4%3
8%2
x)
4x
1 = 0. For this equation to have a solution
(1 > 0, A =1= 1) under the conditions (2), the following inequalities must be fulfilled:
b, c, and

+

+

4%3

f

T

+
D

8x2

=

+

X

>

0,

(4x3 + 8xl

= (2%+1)2 (%+1)

+ +
(3)

+

x 2)

(4%

-

+ 1)2

(2%-1) (2%1+5%+1)

>

0,

(4)

where D is the discriminant of the quadratic equa..
tion. The system of inequalities (2), (3), (4) is
true for

_.!. < x < V17-5
4

4

291


Thus, the original triangle is not necessarily
isosceles. But it has heen proved that it can be isosceles if one of the angles of the original triangle is
obtuse and its cosine lies in the interval

V ~-

(-1 '

5) , which corresponds approximately to an

angle from 102°40' to 104°28' If x = -1/4, then
the constructed triangle degenerates; for x =

V17-5

0

4
we have: LAIBlC = LA l CI B = 90 ,that
is, the two cases considered at the beginning of the
solution coincide for this size of the angle.
204. Let M denote the point of intersection of
AD and KL:

IKMI
IMLI

SAKD

1
2

= SALD = ~

IAKI·IADI sin L KAD
IDLI.IADsin LADL

IAKI·ICDI
IDLI·IAFI •

-

(We have used the fact that the sines of the inscribed
angles are proportional to the chords.) Analogously, if M 1 is the point of intersection of BE

IKMII

IBKI·IEFI

and KL, then we get: IMILI = ILEI.IHel •
But from the similarity of the triangles A KF and
IAKI
BKC, and CLD and FLE, we have IAFI

I CDI
IDLI

IFEI.
= (LET ,

= IBKI
IBCI

'

mu ltiip I·
ylng th ese equ altti
lies,

IKMI
IKM11
.
we get: IMLI
IMILI' that IS, M and M 1
coincide. Remark. We can show that the statement of the problem is retained if A, B, C, D, E,
and F are six arbitrary points on the circle. Usual19*

292


ly, Pascal's theorem is formulated as follows: if
A, B, C, D, E, F are points on a circle, then the
three intersection points of pairs of straight Jines
AB and DE, BC and EF, CD and FA lie on a
straight line.
205. Let N be the point of intersection of the
straight line A tA 1 and the circle, N being distinct
from A s. Apply Pascal's theorem to the hexagon
ABCCsNA s which is possibly self-intersecting
(Problem 204 in Sec. 2)_ Intersection points of
two pairs of straight lines AB and CIN, Be and
NA 2 (the point AI), CC I and AA I (tho point AI)
lie on one straight line. Consequently, A Band
C"N intersect at a point Ct.
206. Let the given mutually perpendicular
straight lines be the e- and y-axes of a rectangular
coordinate system. Then the altitudes of the triangle lie on the lines y = k,z (; = 1, 2, 3)i ill
this ease the sides or the triangle must have slopes

equal to

-f"

and given

the condition

that

the vertices (zC' Yt) belong to the altitudes we find
the ratios of absolute terms
in the equations of
the sides kiY+Z=Ci: cl=ktYs+za, c,,=k.y.+z.,
Cl
k tk.+1
W· h
YI=kaza~
k
,ete. It a properly
1ka+1
chosen unit of length, we may take Cl= k~,
where k = k1k"ka- The points of intersection of the

c,

c;-=

'

k~kl

line kiY+z=

with axes:

(0,

k~kl)

,0) , the midpoint (Pt> of the line
ki
1)
segment between them: ( 2 (k+k,) '2(k+kt) •
and

(k~kl

The slope of the straight line PIP! is equal to
(

2(k~kl)

-

2(k~kl)

) -;- (

2(k~k)1

)
kt
2 (k+ k ) = (k l

293

.

k,,) -:- (kk 2 - kk 1) = -

k1

The
slopes of the lines P"Ps and PsP t are [ust the
same. Therefore the points Pit P", P s lie on
a straight line (its equation: ky x = 1/2).
Remark t. Joining the point H of intersection
of the altitudes of the triangle to the points P t , P 2'
and P 3 with straight lines, we get an intersecting
consequence. Let at, ex , and as be the angles of
the triangle enumeratet anticlockwise, at, a2' and
as the straight lines containing the sides opposite
these angles; three straight lines Ph P2' and Ps pass
through the point H so that the angles between the
pairs P2 and Pa, Ps and Ph PI and Ps (measured
anticlockwise) are equal to at, (%2' as. Then the
points of intersection of the pairs Pt and at, P2
and at, Ps and asHe on a straight line. The particular cases of this theorem are left to the reader
(many of these geometrical facts being elegant,
and far from obvious).
Remark 2. In our problem, instead of the midpoints of the line segments cut out on the sides of
the triangle, we might have taken the points dividing them in the same ratios. These points will
also turn out to be collinear.
207. To determ ine the angles of the triangle
AtBtC h take advantage of the fact that the points
P, A), BIt and CI lie on a circle (the same is true
for the other fours of points). I f the point Plies
inside the triangle ABC, then LAtC1B 1 =
LA 2C 2B 2 = LAPB - LACB. For a scalene triangle ABC there exist eight distinct points P
such that the corresponding triangles AtBtC I and
AsB.C" are similar to the triangle ABC (the triangle
A "B 2C2 being congruent to it) . Of these eight
points, six lie inside the circle circumscribed about
the triangle ABC, and two outside it.
208. The straight lines under consideration are
the middle perpendiculars to the sides of the triangle AlBIC!.
t

-

+

294


209. Notation: ABC is the given triangle, M
the point situated at a distance d from the centre
of the circle circumscribed about the triangle ABC,
A I' B l' and C 1 the feet of the perpendiculars dropped from M on BC, CA, and AB; At. B 2 , C 2 the
intersection points of AM, BM, CM with the
circle circumscribed about the triangle ABC, respectively, a, b, and c the sides of the triangle
ABC, ai- b h Cl and at, b i , C 2 the sides of the triangles AlBICt and A~BiC2' respectively; S, Sit
and 8,. the areas of those triangles, respectively.
We have:
al=IAMI sinA=IAMI

2~'

(1)

The sides bl and CI are found in a similar way.
From the similarity of the triangles B 2MC 2 and
BMC, we get:
at
IBIMI
a

(2)

ICMI

Analogous ratios are obtained for

~2

and

:2

The triangles AlBIC] and A tB 2C,. are similar (see
Problem 207 of Sec. 2); in addition,
82

a l b2c?

(3)

s=libC
Bearing all this in mind, we have:

( ~ ) 3 ==~.
S
81

=(_1_)3
4R2

S~ = aibtc~ • a~b~c~
alb~cl
a3 b3c3
IAMIIIBMI'ICMI 2 a2 b2 cl
83

= ( 4~2
X IB 2M I

ICMI

a3 b3 c3

• a2b2c2

)3 IAMI 2 1BM I21CMI2

IC2MI IAIMI
IAMI
IBMI

(_1_IRI_d21)3
4R2


295

(In the second equality we have used the similarity
of the triangles AlB 1 C1 and A 'l,.B 2 C 2 and the equality (3), in the third the formulas (1), in the fourth
the formulas (2).) Remark. For d = R the area of
the triangle formed by the feet of the perpendiculars turns out to be equal to zero, that is, these
feet are situated on a straight line. This line is
Simson's line (see Problem 153 in Sec. 2).
210. The statement follows from a more general
fact: if on the sides of the triangle circles are constructed so that their arcs located outside the
triangle are totally equal to 4n or 2n, then those
circles have a common point (in our case, as such
a triangle, we may take the triangle with vertices
at the midpoints of the sides of the triangle ABC
and prove that the three circles passing through
the midpoints of AB, AC, and AD; BA, BC, and
BD; CA, CB, and CD have a common point).
211. The statement is based on the following
fact. Let an arbitrary circle intersect the sides
of the angle with vertex N at points A, Band C,
D; the perpendiculars erected at the points A and
D to the sides of the angle intersect at a point K,
and the perpendiculars erected , at the points B
and C intersect at a point L. Then the straight
lines N K and N L are symmetric with respect to
the bisector of this angle. Indeed, LANK =
L.ADK (the points A, K, D, and N lying on the
same circle). In similar fashion, LLNC = LLBC.
Then LADK = 90° - LADN=90o-LNBC =
L LBC. (The quadrilateral ABCD was supposed
to be non-self-intersecting.)
212. Let A, B, C, and D be the given points, D 1
the point of intersection of the straight lines
which are symmetric to AD, BD, and CD with
respect to the corresponding angle bisectors of the
triangle ABC. It was proved in the preceding
problem that the pedal circles of the points D
and D I with respect to the triangle ABC coincide.
Let the straight lines symmetric to BA, CA, and
DA with respect to the corresponding angle bisec-

296

Problems in Plaoe Geometry

tors of the triangle BCD intersect at a point A,.
It is easy to prove that the points Al and D 1 are
symmetric with respect to the straight line CB.
Consequently, the pedal circles of the points D
(or D I ) with respect to the triangle ABC as well
as the points A (or A 1) with respect to the triangle
BCD pass through, the midpoint of DIAl. On having determined the points B 1 and CI in a similar
way, we see that each of the pedal circles under
consideration passes through the midpoints of the
corresponding line segments joining the points
AI' B I , Cl , and D I • Thus, our problem bas been
reduced to Problem 210 of Sec. 2.
213. Let B I and C1 be the points diametrically
opposite to the points Band C, M the second
point of intersection of B ...B I and the circle circumscribed about the triangle ABC, Ci the point of
intersection of AB and CIM. By Pascal's theorem
in Problem 204 in Sec. 2 applied to the hexagon
AB1CMBC" the points 0 (the centre of the circle),
B 1 and C; lie on one straight line, that is, Ci
coincides with Ct. But L.BMH I = LBMB I = 90°,
LCMC1 = LCMC, = 90°; hence, M is one of
the intersection points of the circles with the diameters BB I and ce l • Let N be the second point of
intersection of those circles. Their common chord
M N contains the point H of intersection of the
altitudes of the triangle ABC (Problem 19 in
Sec. 2). If BB o is the altitude of the triangle ABC,
then I MH 1·1 HN I = I BH 1·1 HB o I. Hence (see
Problem 164 in Sec. 2), N lies on the nine-point
circle of the triangle ABC.
218. Let the radius of the circle be r, and the
angles between the neighbouring radii drawn to
the points of tangency, in the order of traverse, are
equal to 2a, 2p, 2y, 26 (a
P y ~ = n).
Then
S = r 2 (tan (X
tan P + tan y + tan 6).
(1)
The sides of the quadrilateral (we are going to find
tan p) =
one of them) arc equal to r (tan (X

+ + +

+

+

297

+

sin (a+~)
·
A)
r cos a cos ~ an d so f or th - Since SID (ex
t' =
sin (1' 6), sin (~
1') = sin (a
6), the formula given in the hypothesis is reduced to

+

+

+

<P+V) sin (1'+ a)
cos ex cos p cos y cos .,

S =r" sin (a+~) sin

(2)

It remains to prove the equality or the right-hand
members of (1) and (2) provided that ex
p + 'V +
6 = rt.
219. Prove that S BNA = S BMC
SAMDo If

+

+

IAMI

ICNI
IABI = INDI =A, then S BMC= (i-A) S BAC'

S AMD = ASBiD- On the other hand, denoting
th.e distances rom C, D, and N by hi' hI' and h,
respectively, we find that h = Ah1
(t - A,) h 2 t
1
Consequently, SABN=TIABI-h=)., 2"IABlh1 +

+

t

(t-'A)TIABlht='ASABD+(1- A)SBAC =
SAMD

+ SBMC·

221. The angles between the sides and also
between the sides and diagonals of the quadrilateral Q I are expressed in terms of the angles between
the sides and between the sides and diagonals of the
quadrilateral Ql- (The diagonals of the quadrilateral Q2 are perpendicular to the corresponding diagonals of the quadrilateral Ql and pass through their
midpoints.)
222_ Consider the parallelograms ABMK and
DCML and prove that K~ divides DA in the same
ratio as the point N, and the straight line M N is
the bisector of the angle KML.
223. First of all, prove that the diagonals of the
gi ven quadrilateral are bisected by the point of
intersection, that is, that the quadrilateral is a parallelogram. Let ABeD be the given quadrilateral,
o the point of intersection of the diagonals. Suppose
that I BO I < I OD I, I AD I ~ I oc I; consider


298

the triangle OA1B 1 symmetric to the triangle OAB
with respect to the point 0; obviously, the radius
of the circle inscribed in the triangle OA IBI is less
than the radius of the circle inscribed in the triangle OCD, while, by hypothesis, they are equal.
Thus, 0 is the midpoint of both diagonals. We
prove that all the sides of the quadrilateral are
equal. We use the formula 8 = pr (8 the area, p
the semi perimeter, r the radius of the circle inscribed in the triangle). Since the areas and the
radii of the circles inscribed in the triangles ABO
and BOC are equal, their perimeters are also equal,
that is, I A B I = I B C I.
224. Using the solution of the preceding problem, prove that the diagonals of the quadrilateral
are bisected by the point of their intersection.
225. The hypothesis implies that ABCD
(Fig. 38) is a convex quadrilateral. Consider the

D
~------+~C

A,
Fig. 38

parallelogram ACCIA I in which the sides AA I and
eCI are equal to each other and parallel to the
diagonal BD. The triangles ADA I , CDC tt and
C.DA. are congruent to the triangles ABD, BCD.
and ABC, respectively. Consequently, the line
segments joining D to the vertices A, C, C1,


299

and A t separate the parallelogram into four triangles in which the radii of the inscribed circles
are equal. If 0 is the intersection point of the
diagonals of the parallelogram ACC/A t , then D
must coincide with 0 (for instance, i D is inside
the triangle COCl' then the radius of the circle
inscribed in the triangle ADA I is greater than the
radius of the circle inscribed in the triangle AOA l ,
and the more so in the triangle CDC ). Thus,
ABCD is a parallelogram, but, in ad JItion, it
follows from Problem 223 of Sec. 2 that ACCtA I
is a rhombus, that is, A BCD is a rectangle.
226. The necessary and sufficient condition for
all four items to he fulfilled is the equality
lAB 1·1 CD I = I AD 1·1 BC I. For Items (a) and
(b) it follows from the theorem on the bisector of an
interior angle of a triangle, for Items (c) and (d)
from the result of Problem 234 of Sec. 1.
227. Let ABCD be the given quadrilateral. We
assume that the angles A and D are obtuse, Band
p

A

....--A-7

a

Fig. 39
C are acute. Denote the feet of the perpendiculars
dropped from the vertex A by M and N, and
from the vertex C by K and L (Fig. 39, a), R the
point of intersection of M Nand LK. Note that
the points A, K, N, C, L, and M lie on one and
the same circle of diameter A C. Let us show that

300

MK


II LN:

LMKL

=

L KCB = L KLN. Thus,

L.MAL=900-LB=

IMRI
IMKI
lRNT = ILNI =

+
+

sin L M CK
sin (L C
L B - 90°)
sin L LAN
sin (L A
L. B - 90°)
cos (LA-LB)
sin (LA+LB-900) Let now P and Q be the feet
of the perpendiculars dropped from the vertex B,
and S is the point of intersection of M Nand PQ
(Pig. 39, b). Since L PNB = L PAB = L C, PN
is parallel to DC, that is, MQN P is 8 trapezoid
(A N BP is an inscribed quadrilateral with

.

diameter

A B) ·

Thus

IABt cos(LA+LD-1800)

IMSI
IA/QI
lSNI = lPNI

==:

cOS(LA- LB)

I AB I sin (L B + LA - 90°) sin (L A + L B - 90°) •

(We have used the fact that MQ is the projection
of AB on DC; the angle between AB and DC is
equal to LA+ LD-180o.) Thus the points Rand
S divide M N in the same ratio, that is, they
coincide; hence, the three straight lines intersect
at one point. Now, it is easy to show that all the
four straight lines intersect at the same point.
228. Let us find the ratio in which BC divides
M N. This ratio is equal to the ratio SSMCB ~
IAICI cos LBCD

I BN I cos L CBA

CBN

..
.
Analogously, the ratio In which

lAM cos LBAD
AD divid es MN·IS equaI to INDIt cos L.ADC •
IV}
But these ratios are equal to each other since
LBCD = LBAD, LCBA = LCDA, and the
triangle AMC is similar to the triangle DNB.
229. Take M 1 such that BCMM I is a parallelogram; M 1 lies on the circle passing through the
points B, M, and A. Since I AMI I = I DM I
(ADMM J.. is also a parallelogram), the triangles
CDM and BA M 1 are congruent, that is, the radius

301


of the circle circumscribed about /the triangle

CDM is equal to R. The radius of the circl~,~jrcums

cribed about the triangle ADM is also equal to R.
230. Let K and L denote the points of tangency
of the given circle with the straight lines AB and
AD. Let, for definiteness, K and-L be situated
inside the line segments AB and AD. On the
straight line CB, we take a point P such that
IBPI = IBKI, B lying betweenPandC,andon
the line CD a point Q such that I DQ I = I DL I,
D lying between C and Q. We have: I CP I
I CB I I BK I = I CB I I AB I - I AK I
I CQ I. The circle passing through the points
P and Q and touching the lines CB and CD
intersects BD at such points M 1 and N 1 for which
the equalities I BM11·' BN 1 1 = 'BM 1·1 BN I;
I CN 1·1 CM 1 1= 1CN 1·1 CM I are valid. These
equa ities imply that M 1 and N 1 must coincide
with M and N,. respectively. The other cases of
arrangement of the points are considered much in
the same way. It is possible to avoid looking over
alternate versions by specifying positive directions
on the lines AB, BC, CD, and DA and considering
directed segments on these lines.
231. For definiteness, we assume that the
points Band D lie inside the circle. Let P and Q
denote the points of intersection of the straight
line B D and the circle (P is the nearest to B), L
the point of intersection of CB and the circle, l
the tangent to the circle passing through the
point c.
Consider the triangle PCN from whose ver...
~ices the straight lines PQ, NM, and I emanate.
With the aid of Ceva's theorem (Problem 44 of
$ee. 2) reasoning in the same way as ;n Problem 49
of See. 2, we get that for the lines PQ, N M, and l
to intersect at one point, it is necessary and sufftcient that the following equality be fulfilled:

+

+

t

IPMI
ICQI
fMCI • IQNI

INCI

.1CPI=1.

(1)

302


On the other hand, in the hexagon ALPMCQ
the diagonals AM, LC, and PQ intersect at one
point. Hence (see Problem 49 in Sec. 2)
IAL 1·1 PM 1·1 CQ 1=1 LP 1·1 Me 1·1 QA I. (2)
Obviously, I NC I = I AL I, I QN I = I LP I,
I CP I = I QA I. Thus, from the validity of the
equality (2) there follows the validity of the
equality (1).
232. i. Since 0 1 is the centre of the circle inscribed in the triangle ABC, we have: LBOIA =
1
90° + 2 L BCA (Problem 46 of Sec. 1). Hence,
LBOIA = LB0 4A, and AB0 104 is an inscribed
quadrilateral (see Fig. 40, a); consequently, the
angle adjacent to the angle BOlO 4 is equal to
L.BAO, =

~

L.BAD. Similarly, the angle adja-

cent to L.B010" is equal

~

to

~

L.BCD. But

(L.BAD+L.BCD) =90°; hence, 0,° 1°,,=90°,

2. To prove the second part of the statement,
let us first show that the distance from a vertex of
the triangle to the point of intersection of the altitudes is completely determined by the size of the
angle at this vertex and the length of the opposite
side, namely (Fig. 40, b): I en I = I cs I X
cos a
IABI
.
. L CAB = - .-cos a, = I AB I cot ct. Since
SIn
SID a,
ABeD is an inscribed quadrilateral, I AHa I =
I BN s I and A H a is parallel to BH,,; hence,
ABHIR s is a parallelogram. Thus, the point of
intersection of AN 2 and BH a bisects these line
segments. Considering the other parallelograms, we
see that the line segments H sA, H aB, H 4C, and
HID intersect at the same point (M) and are bisected by this point, that is, the quadrilaterals
ABeD and H 1 H"H sH 4 are centrally symmetric
with respect to the point M (Fig. 40, c).


303

304


233. If the sides of the triangle ABC, opposite
the vertices A, B, and C, are respectively equal to
a, b, and e, and the angles ADB, BDC, and CDA
are, respectively, equal to a, ~, and y (we assume
that a
p y = 2n), then the distances from
the point D to the intersection points of the altitudes of the triangles ADB, BDC, and CDA are
equal to the magnitudes of c cot a, a cot p, b cot y,
respectively (see the solution of Problem 232 of
Sec. 2). It is easy to make sure that the area of
the triangle with vertices at the intersection points
of the altitudes of the triangles ADB, BDC, and

+ +

CDA is equal to

~

cot a-a cot

~ sin B

+

~

X

cot V sin C
~ b cot V-ccot a sin A =
S ABC (cot ex cot ~+cot ~cot 1'+ cot ycot ex) = S ABC
since the expression in the parentheses is equal to
1. (Prove this taking into account that a + p
l' = 2n). Analogously, we consider other cases of
location of the point D (when one of the angles a, p,
y is equal to the sum of two others).
234. (a) Let ABeD be the given quadrilateral, R
and Q the points of tangency of the circles inscribed
in the triangles ABC and ACD, respectively,
with the straight line A C. Then (see Problem 18
a cot

~-b

c

+

+

of Sec. 1) I RQ I = II AQ I-I AR II =

I A C I - I BC

I) -

1

2" I (lAB I +

n AD I + I A C I -

I CD I) I =

~ IIABI+ICDI-IADI-IBCII_ Since ABCD
is a circumscribed quadrilateral, lAB I ICD I =
I AD I + I Be It that is, I RQ I = o.
(b) If K, L, M t N are the points of tangency of
the circle with the sides of the quadrilateral, and
Kit Lit M 1 , and N I the points of tangency of the
circles inscribed in the triangles ABC and ACD
(Fig. 41), then NIK! II NK, and MILl II ML. Let
us prove that K1L J II KL and NIM! II NM. Since
the circles inscribed in the triangles ACB and A CD

+


305

touch each other on the diagonal at a point P, we
have: I ANt I == I AP I :=: I AM I,
that
is,
N 1M 1 II NM. Consequently, KtL1M1N 1, as well as
K LMN, is an inscribed quadrilateral.

C

o

Fig. 41
235. Let 01' 0,., 03' 0 4 denote the centres of
the circles inscribed in the triangles ABC, BCD,

A

A
D

Fig. 42
CDA, and DAB, respectively, (Fig. 42, a, b). Since
0 1020304 is a rectangle (see Problem 232 in Sec. 2),
we have: I °103 I == I ° 2°4 I. If K and L are the
20-01557

D

306


points of tangency with A C of the circles inscribed
in the triangles ABC and ACD, then I KL I =
1
"2 I IAB I I CDI-IBCI-tADII (see the solution of Problem 234 in Sec. 2). Analogously, if
P and Q are the points of tangency of the corresponding circles with BD, then I PQ I = I K L I.
Through 03' we draw a straight line parallel to A C
to intersect the extension of 0lK. We get the
triangle 0lOaM; we then construct the triangle
O,,04R in a similar way. These two right triangles
are congruent, since in them: 10 10 3 I = 10,,0 4 I,
I03 M I = IKLt = IPQI == I04RI. Hence, IO.M.I=
I O"R I; but ·1 0IM I equals the sum of the radii
of the circles inscribed in the triangles A Be
and ACD, and I 02R I is equal to the sum of
the radii of the circles inscribed in the triangles
ACD and BDA (see also Problem 315 in Sec. 2).

+

C

...----+--------IIll

A
Fig. 43
236. In the quadrilateral ABeD (Fig. 43):
a, I Be I == b, I CD 1 == c, I DA I === d,
m, I BD ~ == n, We construct externally

r AB I =
I A C I ==

307

on the side A B a triangle A KB similar to the
triangle ACD, where LBAK == LDCA, LABK ==
L CAD, and on the side AD we construct the triangle AMD similar to the triangle ABC, where
L.DAM = LBCA, LADM == LCAB. From the
corresponding similarity we get: I A K I == ~ ,
m

I AM I == ~, IKBI == IDMI ==~. In addim
m

+

+

+

LKBD
L.MDB == LeAD
LABD
L.BDA
LCAB == 180°, that is, the quadrilateral KBDM is a parallelogram. Hence, I KM I ==
I BD 1== n, But LKAM = LA + LC. By the
law of cosines for the triangle KAM, we have:

tion,

+

n2= (': )2+(~)2_2(::) (~)COS(A+C).
whence m 2n2 == a2c2
b2d2 - 2abcd cos (A
C).
237. The statement of Ptolemy's theorem is a
corollary of Bretschneider's theorem (see Problem 236 of Sec. 2), since for an inscribed quadrilateral LA
LC == 180°.
238. If M B is the greatest of the line segments
I MA I, I MB I, and I MC I, then, applying Bretschneider's theorem (Problem 236 of Sec. 2) to the
quadrilateral ABCM, we get: IMBI 2 = IMA II
2
I MC 1 _ 2) MA I·) MC ) cos (LAMC+600), that
is, IMBI < IMA I IMCI since LAMe =#= 120°.
239. Replacing in the expression

+

+

+

+

+

ta,flt v6

+

t~vt6a.

==

ta.vtafJ

(1)

the segments of the tangents with the aid of the
formulas obtained when solving Problem 201 of
Sec. 1, we make sure that if the relationship (t) is
fulfilled for some circles a, ~, "i, and 6 touching
the given circle at points A, B, C, and D, then
it is fulfilled for any such circles. It remains to
check the validity of the relationship (t) for some
particular case. If a, ~, y, and 6 are circles of
zero radii, then we ,et an ordinary Ptolemy's theorem (Problem 2370 Sec. 2). In order not to refer to
20*

308


Ptolemy's theorem, we may take the circles a and
() having a zero radius, and the circles 6 and ~
touching both the circle circumscribed about the
quadrilateral ABeD and the chord AD. In this
case, the validity of the relationship (1) is readily
verified. Hence, in accordance with the remark
made, we get the validity of (1) in all the cases
(thereby we have simultaneously proved Ptolemy's
theorem itself) .
240. When proving our statement, we shall use
the method of "extension" of circles. The essence
of this method consists in the following. Let two
circles, say a. and ~, touch externally some circle
1:. Consider the circles a't P't and ~' which are
concentric with a, P, and 1:, respectively. If the
radius of the circle t' is greater than the radius of
the circle 1: by a quantity X and the radii of the
circles a' and P' are less than those of the circles ex
and 6 by the same quantity Xwhich is sufficiently
small, then the circles a' and ~' touch the
circle 1:' externally, and the length of the common
external tangent to the circles a' and P' is equal
to the length of the common external tanBent to
the circles ex and p. The case when a and p touch
the circle 1: internally is considered in the same
way. And if one of the circles a and 6 touches 1:
externally, and the other internally, then, with an
increase in the radius of ~, the radius of the first
circle decreases and the radius of the second circle
increases, the length of the common internal tangent to the circles a I and P' remaining unchanged.
For the sake of definiteness, consider the case
when in the equality (.) (see the statement of the
problem) there appear only the segments of the
common external tangents. (Note that none of the
circles can be found inside the other.) Let us prove
that the circles a, p, y, and 6 touch a certain circle
1: in the same manner, all of them either externally
or internally. Let not all of the circles a, p, 'V, and
6 have equal radii (the case of equal radii is readily considered separately), and, for definiteness,


309

let Tcz , the radius of the circle a, be the smallest.
Consider the circles a', p', »', 6', where a' is a
circle of zero radius, that is, a point coinciding
with the centre of the circle a and ~', y', ~' circles
concentric with the circles 6, 'V, 6 with radii reduced by the quantity Ta,. For further reasoning, let
us take advantage of the following assertion which
is marked by (T):
If P', 'V't 6' are three circles none of which lies
inside another and at least one of them has a nonzero radius, then there are exactly two circles 1:1
and 1: 2 each of which touches the circles ~', "/',
and 6' in the same manner. We shall return to
this assertion at the end of the solution.
On the circles II and ~2' take points ~l and
t~d~'

t~2P'

fa,p'

t~16'

ta.26'

ta,'tJ'

a2 such that ---=---=.--=A.,

(Xl

and

a lying on the arcs not containing the point
o tangency of the circle 1'. For three fours of
circles (a', ~', ,,/',6'), (CXIt p', 'V', 6'), (a~" ~', s'.
6') the relationship (.) is fulfilled: for the first
four circles, this is the assertion of the problem,
for two other fours-on the basis of the assertion of
Problem 239 of Sec. 2 (a', (Itt ex! are circles of
t p'
ta, e:
zero radius). Consequently, ~ =
t a 2,"
ta.21"

l

__
2_=

'a,'p'
--=J.1.
ta.'v'

But the locus of points M for which the ratio of
tangents to two fixed circles is constant is a circle
(see Problem tt in Sec. 1). Hence, a tt (XI' and a'
belong both to the locus of points for which the
ratio of the tangents drawn to the circles p' and
6' is equal to '" and to the locus of points for which
the ratio of the tangents drawn to the circles ~' and
V' is equal to p:. And this means that a' must coincide either with al or a 2 •
Let al and (X2 coincide. Prove that in this case


310

the circles defined by the parameters A and p. touch
each other. Let us take A =1= A, but sufficiently close
to A. Then 1 defines on ~1 and ~ 2 two points ~1
~
and a 2 for which
~

t~~,

t;~,

tal'"

t'; ~'
t~
~
_1_= __~' =A. We find:
2_
t~lO'

t~20'

C£2'"

f! = --;:}-=-t~2

. Hence, the circles correspond-

ing to the parameters rand ; have a common
A, then ~ -.. p., I ~lal I -+ 0,
chord ~1~2- If
that is, the circles corresponding to the parameters
At andu touch each other at a point at = as_ Thus,
a', p', t', and 6' touch either ~ t or ~ s- "Extending" ~ 1 or ~ 2 by the quantity ±ra" we get that
a, p, y, and 6 touch a circle or a straight line
(1: 1 or I 2 may turn out to be a straight line) or
If in the equality (.) some of the line segments
are segments of common internal tangents, then
we have to prove the existence of a circle :E touching a, p, y, and 6 and such that those of the
circles a, ~, y, 6 for which in the equality (.) there
appears a common internal tangent touch ~ in
different ways. The assertion (T) must change accordingly.
Let us return to the assertion (T). By means of
"extension", we can reduce the assertion to the
case when one of the circles P', v', and ()' has a zero
radius, i ,e, is a point. The reader familiar with the
notion of inversion can easily prove that the assertion (T) now turns out to be equivalent to the
assertion that any two circles not lying one inside
the other have exactly two common external tangents (see Appendix). Remark. If three of the four
given circles a, p, y, () have a zero radius (they are
points), the proof can be considerably simplified.

r -..


311

Do this independently. Henceforward (see Problem 287 ·of Sec. 2), we shall need just this particular case.
24 t. Show that each of these condi tions is both
necessary and sufficient for a circle inscribed in
the quadrilateral A BCD to exist (see also Problem 19 in Sec. 1).
242. Show that each of these conditions is both
necessary and sufficient for a circle, touching the
lines AR, BC, CD, and DA, whose centre is outside the quadrilateral A BCD, to exist.
243. Let A BCD be a circumscribed quadrilateral, 0 the centre of the inscribed circle, M t the
midpoint of AC, M 2 the midpoint of BD, r the
radius of the circle (the distances from 0 to the
sides are equal to reach), Xl' Yl' %1' and Ul the
distances from M 1 to AB, BC, CD, DA, respectively; %2' Y2' %2' and U 2 the distances from M 2 to
the same sides, respectively. Since I A B I
I CD I = I BC I I DA I, we have: I AB I r I BC I r
I CD I r - I DA I r = O. In addition,

+

+

+

I AB I Xl - I BC I Yt + I CD I %1 - 1DA I Ul ==
0, I AB I %2=1 Be I Y2+ I CD 1%2==1 DA I U 2 =

0, and this just means that the points 0, M l'
and M 2 lie on a straight line (see the remark to
Problem 22 of Sec. 2). Other cases of the arrangement of the points A, B, C, and D and the centre
of the circle are considered exactly in the same
way. Here, use the relationships occurring among
the line segments I AB I, I Be I, I CD I, I DA I
(see Problems 241 and 242 in Sec. 2), and, as is
said in the remark to Problem 22 of Sec. 2 assign
unlike signs to corresponding distances if any two
points turn out to be located on both sides of a
straight line.
244. Let Land P denote the points of intersection of the straight lines A M and A N with the
circle, respectively. As there follows from Problem 204 of Sec. 2, the straight lines B L, D P, and
M N meet at one point. But, being diameters, BL
and D P intersect at the centre of the circle, conse-


312

quently, M N passes through the centre of the
circle.
245. Make use of Pascal's theorem (Problem 204
in Sec. 2).
246. Let P denote the point of intersection of
the diagonals, and K, L, M, and N the feet of the
perpendiculars from P on AB, Be, CD, and DA,
respectively, (Fig. 44). Since P KBL is an inscribed

A

Fig. 44
quadrilateral, we have: L PAL = L PBC, analogously, LPKN = LPAD; but LPBC = LPAD
since they are subtended by the same arc. Consequently, K P is the bisector of the angle N K L;
hence, the bisectors of the angles of the quadrilateral KLMN intersect at the point P which is just
the centre of the circle inscribed in the quadrilateral KLMN. Let now AC and BD be mutually
perpendicular, R the radius of the given circle, d
the distance from P to its centre, I A P 1·1 PC I =
R2 -

d2 •

The radius r of the sought-for circle is equal, in
particular, to the distance from P to KL. Denoting L.KLP = L.ABP = a, LPBC = P, we find:


r= IPL I sin ex. = IPBI sin ~ sin ex. = IPBI
~_

IABI -(R

sin (a+p>

IACI
Answer:

2_

2

d)

SiS

:=~:

X

IPBIIACI
IABI sin (a.+~) X
2SABC • _t_ = Rt-d 2
2SABC
2R
2R

IRGI

(R2_d 2)
R2~dl

2R

247. Let ABCD be the given quadrilateral, P the
point of intersection of the diagonals, K the midpoint of BC, L the midpoint of AD (Fig. 45). Let

Fig. 45
us prove that the straight line LP is perpendicular
to BC. Denoting the point of intersection of LP
and BC by M, we have: LBPM = LLPD =
LADP = LPCB. Consequently, PM is perpendicular to BC. Hence, OK is parallel to LP.
Similarly, P K is parallel to LO, and KOLP is
a parallelogram, I LK 12 + I PO )2 = 2 (I LP 12
I PK Ie) = 2 ( IAfie
IBfle ) = 2Re. (If the

+

+

chords AD and Be are brought to a position in

314


which they have a common end point and the corresponding arcs continue each other, then a right
triangle is formed with legs I AD I and I BC I and
hypotenuse 2R, hence, I AD r~
I BC 12 = 4R2.)
Consequently, I LK 12 = 2R2 - dJ, and the points
Land K lie on the circle with centre at S (the midpoint of PO) and the radius t/2 V2R2_tP. But
LM K is a right triangle, M S is its median, 1M S 1=

+

~ I LK 1= -} V2R2 -,p,

that is, M lies on the

same circle.

Answer: 1/2 Y2R2 - tJ2.
248. From Problems 246 and 247 it follows
that if the diagonals of the inscribed quadrilateral
are mutually perpendicular, then the projections of
the intersection point of the diagonals of this
quadrilateral on its sides serve as vertices of a
quadrilateral which can be inscribed ina circle and
about which a circle can be circumscribed. The
radii of the inscribed and circumscribed circles and
the distance between their centres are completely
determined by the radius of the circle circumscribed
about the original quadrilateral and the distance
from its centre to the intersection point of the diagonals of the quadrilateral inscribed in it. Consequently, when the diagonals of the original quadrilateral are rotated about the point of their intersection, the quadrilateral formed by the projections
of this point rotates remaining inscribed in one and
the same circle and circumscribed about one and
the same circle. Taking into consideration the expressions for the radii of the inscribed and circumscribed circles obtained in the two previous problems, it is easy to show that the relationship to he
proved is fulfilled for such quadrilaterals.
To complete the proof, it remains to prove that
any "inscribed-clrcumscrtbed" quadrilateral can be
obtained from an inscribed quadrilateral with
mutually perpendicular diagonals using the above
method. Indeed, if KLM N is an "lnscribed-clrcum.


315

scribed" quadrilateral, P the centre of the inscribed
circle, then drawing the lines perpendicular to the
angle bisectors K P, LP, M P, and N P and passing
through the points K, L, M, and N, respectively,
we get the quadrilateral ABeD (see Fig. 44). In

this

case, L.BPK = L.KLB = 90°-+ L.MLK

(here, we have used the fact that in the quadrilateral P KBL the opposite angles are right ones and,
consequently, it is an inscribed quadrilateral). Similarly.
and,

L.KPA

hence,

=

L.KNA

= 90" - ~

LBPA = LBPK

L.MNK,

+ L.KPA

==

t80° - ~ (L. M LK + i. M N K) = 90°. Thus, all
the angles BPA, APD, DPC, and CPB are right
ones, P is the intersection point of the diagonals of
the quadrilateral A BCD, the diagonals themselves
being mutually perpendicular. It is easy to show
that ABCD is an inscribed quadrilateral since

+ LADC = LPBL + LPBK
+ LPDN + LPDM = LPKL + LPLK
+ L.PMN + L.PNM = ~ (L.NKL + L.KLM
LABC

+ LLMN +

L.MNK)

=

180°.

Note:: see also Problem 319, Sec. 2.
249. The midpoints of the sides of the quadrilateral form a parallelogram whose diagonals are
parallel to the line segments joining the centres of
mass of the opposite triangles. The other parallelogram is formed by the four altitudes of the triangles in question emanating from the vertices of the
quadrilateral. The sides of the first parallelogram
are parallel to the diagonals of the quadrilateral,
whileethose of the second parallelogram are perpendicular to them. In addition, the sides of the second parallelogram are cot a times greater than the

316


corresponding sides of the first one «(1, is an acute
angle between the diagonals of the quadrilateral).
250. We prove that both assertions (BD is the
bisector of the angle AN C, and A C is the bisector
of the angle BMD) are equivalent to the equality
lAB 1·1 CD I = I AD 1·1 BC I. On the arc BAD
we take a point Al such that I DA t I = lAB [.
The conditions of the problem imply that the
straight line AIC passes through N, the midpoint of
BD, that is, the areas of the triangles DA lC and
AtBC are equal, whence I DA t 1·1 DC I =
IBAtl·t BC I, that is lAB 1·1 CD I = I AD I X

I BC I.

251. The perpendicularity of the angle bisectors
is proved quite easily. Let us prove the second
assertion. Let M denote the midpoint of AC, and
N the midpoint of RD. From the similarity of the
triangles A KC and BKD, it follows that L M KA =

IMKI

IACt

.

.

LNKD and IKNI = IBDI ' that IS, the bisector of the angle B K C is also the bisector of the
angle M KN and divides the line segment M N in

.

IMKI

IACI

.

the ratio IKNI = IBDt
Obviously,
the
bisector of the angle A LB divides the line segment MN in the same ratio.
252. Let ABCD be the given quadrilateral,
the centre of the circle circumscribed about the
triangle ABC, 0 1 and 0 1 the centres of the circles
circumscribed about the triangles DAB and BCD,
K and L the midpoints of the sides AB and BC.
respectively. The points 0 1 and 02 lie on OK and
100
1002 1
ThO
OL, respectiveI y, an d 101 1 1
101LI.
IS
KI
follows from the fact that 0 10 t is perpendicular to
DB and, consequently, parallel to LK (LK is parallel to A C). Hence, the straight lines A 0t and
CO. divide OB in the same ratio. (We apply Menelaus' theorem (Problem 45 in Sec. 2) to the triangles OKB and OLD.)

°

0

317

Answers. Hints, Solutions

253. Let R denote the radius of the circle, and
b, and c the distances from P, Q, and M to its
c~ntret respectively. Then (Problem 272 of Sec. t)
2
f QP It == a2
b2 - 2RI,
I QM 1 = b2 c'- 2
2R1 , I PM 1 = c2
a2 - 2R2. If 0 is the centre
(J

+

+

+

of the circle, then for QO to be perpendicular to

PM, it is necessary and sufficient that the inequali2
ty I QP p~ - I QM I' = I OP 1 - I OM)2 or
(til

+

b2

-

2R2) -

(bl

+

cl

-

2R2)

=

a2

-

c2

(Problem 1 of Sec. 2). The perpendicularity of the
other line segments is checked in a similar way.
2M. If M, N, P, and Q are the points of taopoey of the sides AB, Be, CD, and DA with the
efrete, respectively, then, as it follows from the
solution of Problem 236 of Sec. 1, M P and N Q
.meet at the point of intersection of A C and BD. In
altoilar fashion, we prove that the lines M Nand
I'Q meet at the point of intersection of the straight
l.lbes AC and KL, and the straight lines MQ and
N P at the point of intersection of the lines KL
and BD. Now, we use the result of the preceding
problem for the quadrilateral M N PQ.
255. Denote: LDAN = LMAB = cpo Let L
be the point of intersection of A M and N B, P the
~iDt of intersection of A Nand D M, Q the point
uf intersection of A K and MN. By Ceva's theorem
(Problem 44 of Sec. 2), for the triangle AMN we
have:

lNQI

IALI

1QMT= ILMI
l~f'IAMI sin
..

cos q>

INPI

e1PAf=

SNAB
SNMB

__
-

L,NAB IANI ' NM 1 tan e cos L,ANM
2

tD/ t INMI tanq> cos LAMN21ANt
""
2

SDNM
• SDAM

cose

IAMI sin LMAD

318

IAN' cos LANM

IAMI cos LAMN '
that is, Q divides N M in the same ratio as the altitude drawn from A on N M .
257. First, prove the following additional assertion: if A, B, and C are collinear points, M is an
arbitrary point in the plane, then the centres of
the circles circumscribed about the triangles MAC,
MBC, MCA and the point M lie on one and the
same circle. Then use the result of Problem 256,
Sec. 2.
258. Let A, B, C, D, P, and Q denote the intersection points of the straight lines (the points are
arranged in the same way as in the solution of
Problem 271 of Sec. t); 0 the centre of the circle
passing through A, B, C, and D; R its radius; a
and b the tangents drawn to the circle from P
and Q, respectively. The fact that M lies on PQ
was proved when we were solving Problem 271 of
Sec. 1. In addition, it was proved that I PM I X
I PQ I

=

a2 ,

V a2 + b2 •

Ya2 +b2

Thus, I PM I
In

Vb2 -

1 QO I =
= a2

2
I QO 1

I QM 1·1 QP I = b2 ,

-

addition
R2.
b2 =

Va la' b2
+
1 PO I

'

I QM I =

= Y a2

Consequently,

2_1
I PM 1

I QP 12 =

QM

-

I PO 12

R2,
-

2
1 • And this

means that OM is perpendicular to PQ. To
complete the proof, we have to consider the case
when (using the same notation) the points A, C, P,
and Q are found on the circle (see also Problem 253
in Sec. 2 and its solution).
259. If one of the straight lines is displaced parallel to itself, then Euler's line of the triangle one
of whose sides is the line displaced moves parallel
to itself. Taking this into account, we can easily


319

reduce the problem to the following. Let A, C,
and D be three collinear points, and B an arbitrary
point in the plane. If Euler's line of the triangle
ABC is parallel to BD, then Euler's line of the
triangle eBD is parallel to AB (Fig. 46). Let us
prove this. We denote: LBCD == q> (we assume
that C lies between A and D, q> ~ 90°), 0 1 and HI
the centre of the circumscribed circle and the intersection point of the altitudes of the triangle
H

Fig. 46
ABC, respectively, 02 and H 2 the centre of circumscribed circle and the intersection point of the
altitudes of the triangle BCD. Describe a circle
about ABHl to intersect 0lHI at a point M. Let us
prove that the quadrilaterals 0IA MBand 02D H 2B
are similar. First of all, the triangles 0IAB and
,OIDB are similar isosceles triangles, and L MAB=
LMH1B = LHIBD = LH 2 BD (BD is parallel to
°IH1 ) , LMBA == LMH1A == LH,DB (AR I and
DH 2 are perpendicular to CB). The similarity of
the quadrilaterals has been proved. Further:
LOIHIB= LOIMA== L.HlMA =L HIBA ==LHIBA,

that is, H 20 2 is parallel to A B .

320


260 _ I t follows from the result of Problem 19
in Sec. 2 that the common chord of the circles with
the diameters AE and DC (and also DC and BF,
B F and A E) contains the intersection points of
the altitudes of the triangles ABC, BDE, DAF,
and CE F ° Further, let K denote the point of intersection of A E and DC and L the point of intersection of AE and BF. By Menelaus' theorem
(Problem 45 in Sec. 2), for the triangles BEA and

.
we have.

EAC

IAKI
IECI
lKET-1CBI·

t,

...
Dividing
these
equalities, one by the other, termwise and bearing
..
ICEI
IBDI
IAFI
In mind that
IEBI
IDAI· IFCI
i, we
IALI

~

.

IEBI
ICFt
°lBCl·1F"'A/ =

IBDI_

IDAI -

IAXI

IKEI

1.

..

get. lALI=l'IEI Consider the circle with
diameter A E. For all points P of this circle the
ratio ~:II is constant (see Problem 9 of Sec. 2).
The same is true for the circles with diameters DC
and B F. Thus, these three circles intersect at two
points PI and Pi such that the ratios of the distances from PI and P t to K, L, and M for them are
equal. Now, we can use the result of Problem t 4,
Sec. 2.
26t. The statement follows from the result of
the preceding problem.
262_ Let I (ABC) denote the midperpendicular
to the line segment joining the point of intersection of the altitudes to the centre of the circle circumscribed about the triangle ABC. Let a straight
line intersect the sides BC t CA and AB of the
triangle A Be at points D, E, and F, respectively.
Let us first prove that as the straight line DE F
displaces parallel to itself, the point M of intersection of the lines l (DFB) and l (DEC) describes
a straight line. Let the points D 1 , E lt F t ; D 2 , E 2 ,

321


F t; D a, E 3' F a correspond to three positions of
this line. The lines l (DiFiB) and I (DiEiC), where
i = 1,2,3, meet at M i and intersect the straight
line BC at points N i and K i . It is easily seen that
the point N 2 divides the line segment N IN 3 in the
same ratio in which the point K 2 divides the line
segment KIKa. This ratio is equal to the ratio in
which D 2 divides D1D a, E,. divides EtE s' and
F ,-F1F a. Since the straight lines l (DiFiB) are
parallel, and the straight lines 1 (D i E iC) are also
parallel, the line l (D 2F2B) divides the line segment M 1M3 in the same ratio as the line 1 (D 2E 2C),
that is, M'I. lies on the line segment M 1 M a.
Let us now show that the point M describes a
straight line I (ABC). To this end, it suffices to
prove that for two positions of the straight line
DEF the corresponding point M lies on 1 (ABC).
Consider the case when this line passes through A
(the points E and F coincide with A). We introduce a coordinate system in which the points A, B,
C, and D have the following coordinates: A (0, a),
B (b, 0), C (c, 0), D (d, 0). We then find the equation of the straight line I (A B C). The intersection
point of the altitudes of the triangle ABC has the

coordinates (0, -

b: ), the centre of the circum-

scribed circle the coordinates (

~ ))

b~- c, ~

(a+

Let us write the equation of the straight

line l (ABC):

z(b+c>+y

3bC)
(a+-a-

Replacing c by d in this equation, we get the equation of the line l (ABD), and replacing b by d the
equation of the line l (A CD).
We can verify that all the three straight lines
have a common point Q (xo, Yo), where xo=
21-01557

322

Fig. 47


323


t

T

(b

+ c + d) -

3bcd
1
~' Yo= 4a (a2-bc-cd-db).

And this is the end of the proof since the case
when the line DEF passes through B or C is
equivalent to the above case.
263. Let I, m, n, and p be the straight lines
which form the triangles (Fig. 47, a). Let us introduce the following notation: P is the centre of the
circle inscribed in the triangle formed by the lines
I, m, and n, and P 1 is the centre of the escribed
circle for the same triangle which touches the side
lying on the line i. The 'notations L, M I" N m ,
etc. have the same sense.

L
M

I
I

N
p

Pm

I

Mp

N,

I

Ln

Ql

Q2

I
I
I

I

Ml

I

Lm

I

Nm

I
I

P,

03

Pn
Np

Lp
Mn

Q4

In the above table, the four points forming a
row or a column lie on the same circle, the centres
of the circles corresponding to the rows lying on
one straight line (ql), while the centres corresponding to the columns on the other (q2); ql and q2
are mutually perpendicular and intersect at Michell's point (Problem 256 in Sec. 2). Let us prove
this. The fact that the indicated fours lie on the
same circle is proved easily. Let 0i' Qf (i ="1, 2, 3,
4) denote the centres of the corresponding circles.
Let us prove that 0 10 2 is perpendicular to QIQ3
and Q2Q4. If in the triangle (l, m, n) the angle
between land m is equal to a, then L LNM l =
a
L LmP M = 90°
'2; consequently, L. L01 M l =

+

21*

324


L Lm0 2M =

1800

-

a.

In

similar

fashion,

LLPmM= LLmP lM l=aI2, LLQ1M = LmQaMl=
a. The triangles L01M l , L m0 2M, LQ.M, LmOaM,

are isosceles ones, their lateral sides being respectively perpendicular (for instance, O.L and LQ1).
Further (Fig. 47, b) IQI0112-IOI0aI2= (a2+c2 ) -

(a2 -f- cJ2)

=

(b2

+c

2

)-

(b2

+ cP) =

102Ql1 2 -

2

I 02Qa1 •

Consequently, 0 10 2 and QIQ3 are mutually perpendicular. In similar fashion, we prove that 0 10 2
and Q204 are also mutually perpendicular (consider
the straight line on which the points N, P, N p»
and P« are located). Therefore QIQ3 and Q2Q4 are
parallel (if these points do not lie on one and the
same straight line) .. In similar fashion, QIQ4 and
QaQ2 are also parallel (they are perpendicular
to °1 0 a), QIQ2 is parallel to QaQ4 (they are
perpendicular to 0 1°.), and this means that
the points Ql' Q2t Q3' Q4 are collinear, they lie
on the straight line qt; the points Ott 02'
0" 0 4 are also collinear, they lie on the line ql.
Obviously, ql and q2 are mutually perpendicular.
We shall displace the straight line m parallel
to itself. Let L', u; 0;, O2 correspond to the
. h I·
'Th
t·
1010;1
ILL'I
straig tine m
e ra 10 101 1
I LmL:mI

°2

is constant (it is equal to I~~II ). This means
that when the line m is displaced, the line 0 1° 2'
that is, ql passes through a fixed point. The straight
line q2 also passes through a fixed point. Since ql
and q2 are mutually perpendicular, the point of
their intersection describes a circle. But when m
passes through A (and also B or C), the points L
and L m coincide with A, the lines 0 10 2 and QIQ 3'
that is, ql and q2 pass through A (correspondingly,
B or C). Thus, the point of intersection of ql and
q2 traverses the circle circumscribed about the
triangle ABC. Displacing the other lines (l, n, p),
we prove that the point of intersection of ql and
q:l belongs to any circle circumscri bed about one of


325

the triangles formed by the lines I, m, n, p , that
is, the lines ql and q2 meet at the intersection point
of the circles circumscribed about those triangles,
that is, at Michell '8 point.
Note that we have proved at the same time
that the four circles circumscribed about the four
triangles formed by four straight lines in the plane
intersect at one point (Problem 256 of Sec. 2).
266. Let C denote one of the intersection points
through which the straight line passes. Let B 1 , B-A'
B 3 be the feet of the perpendiculars dropped respectively from 01' 02' Os on the straight line, and K
and M the points of intersection of the straight
lines, parallel to AlA 2 and passing through 0 1 and
01' with OtBS and OsB s. Since B l and B 2 are the
midpoints of the chords A 1C and CA , we have:
I BIB,. I ~ I AlA 2 1/2. If a is the angle between the

straight lines A1A a and 0lOa, then
21B1B2 1
2 IOl K I
10 0 1 =
10 10 21 == 2 cos a;
in
1

2

~:~::

0 01

SImI ar

.
IA2As i
fashion, 1020s1
2cosa.
268. Let 0 1 and O 2 be the cen tres of the circles,
RJ and R 2 their radii, 1 0 10 2 1 = a, M the point
01 intersection of the common internal tangents.
A circle of diameter 0 102 passes through the points
of intersection of the common external and internal tangents. In the homothetic transformation
with the centre of similitude at the point M and
the ratio of a - R 1 - R 2 this circle goes into the
a
circle with the centre on the straight line 0 10 2
which is tangent to the given circles externally.
269. Let M be one of the points of intersection
of the circles; then MA and MC are the bisectors
of the (exterior and interior) angle M of the triangle BMD since the circle of diameter A C is the
o

•

locus of points M for which

IMAI
IMCI

IMBI
IMDI

326


(see Problem 9 in Sec. 2) Using the relationships
between the angles of the right triangle AMC and
the triangle BMD, make sure that the radii of the
circumscribed circles drawn from the vertex M
are mutually perpendicular.
271. Note (Fig 48, a) that the triangle A PM
is similar to the triangle AMQ, A PL to A K Q,
and AKN to ALN; from these facts of similarity
0

0

we

. IPMI
IMQl

get.

IAMI

= lAQI'

IQKI
(P.L""l

IAQI

=~,

O
ILNI
IALI
INKI == IANI· Mu ItO lvinz theseequa10ties an d
rp ying
I

taking into consideration that I A M I = I A N I,

.

IPMI

IQKI

ILNI

we get that IMQI .~. INKI=1, and this
(see Problem 49 in Sec. 2) is just a necessary and
sufficient condition for the straight lines M N, P K,
and QL to meet in one point.
The method of constructing tangent lines by a
ruler only is clear from Fig. 48, b. The numbers
1, 2,
indicate the succession in which the
lines are drawn,
272. The desired set is a straight line which is
the polar of the poi nt with respect to the given
circle (see Problem 21 in Sec. 2).
273. The angles A M N an d B N M can be expressed in terms of the central angle corresponding to
the arc AB of the given circle (consider various
cases of location of the point N); this done, it is
possible to determine the angle A M BoThe soughtfor locus is a circle.
274. Take advantage of the results of Problems 271 and 21 in Sec. 2. The obtained locus coincides with the locus in Problem 21 of Sec. 2, that
is, this is the polar of the point A with respect to
the given circle.
275. Let 0 denote the point of intersection of
AM and DC (Fig. 49). Through the point B, we
draw a tangent to the second circle and denote the
point of its intersection with A C by K (as in the
0

328


hypothesis). Obviously, the statement of the
problem is equivalent to the assertion that KO is
parallel to CM. Let the angle subtended by the
arc A B in the first circle be (x, in the second p,
then
LBCM == LBAC,
LBDM == LBAD,
LDMC == 1800 - LBDM - LBCM == 1800 0
LBAD - LBAC = 180 - LDAC; consequently, ADMC is an inscribed quadrilateral, L AMC =
p. Further, if the tangent B K intersects D M at a

H

Fig. 49
L, then L KBO = L LBD = L BDL ==
L CA M; hence, KABO is also an inscribed quadrilateral, and L KOA = L KBA == p, that is, KO
point

is parallel to eM (the cases of other relative positions of the points D, B, and C are considered in
similar fashion).
276. Since the circle with diameter CD passes
through a fixed point A on MN (MN 1. CD), the
quantity
I CN 1·1 ND I

==

2
INA 1

(1)

is constant. Denote the point of intersection of PQ

and MN by K. Let us show that
stant. Note that L PNQ = 1800 -

'1;:11

is a con-

L PMQ; hence t

329


IMKI _
IKNI -

-t::,I:

sPMQ __ IPMI·IMQI
SPQN··-IPNI·INQI

IMNI.IMNI __
leNI INDI-

(we have used Equality (1) and

the

fact that the triangle M N P is similar to the
triangle MNC, and the triangle MNQ to the tri8J)gle MND).
277. The equality L.01A02 = L.MAN follows
from the result of Problem 279 of Sec. t , the equality L OI A 0 2 = 2L CAE was proved when solving
Problem 275, Sec. t.
278. Let and 0 1 denote the centres of the two
circles under consideration (0 the midpoint of AB),
K the point of tangency of the circles (K on the
straight line DOt), N the point of contact of the
circle 0 1 with the straight line CD, M the point
of intersection of A B and CD. Since DIN is parallel
to AB, and the triangles K01N and KOA are isosceles and similar, the points K, N, and A are collinear. Let t denote the tangent to the circle 0 1
from the point A (the circle 0 1 is assumed to lie
inside the segment CBD). We have: t 2 = I AN I X
I AK I = I AN 12 + I AN 1·1 NK I == 1 AM 12

°

I MN
(I CM

+

II + ICN 1·1 ND I = lAM 12 + I MN 12 +
I - I MN I) (I CM I + I MN 1)= IAMI 2 +

CM 12 = 1 A K 12 , which was to be proved.
279. Let A be the midpoint of the arc of the
given circle not contained by the segment, and let
the tangents from A to the circles inscribed in the
segment be equal (Problem 278 in Sec. 2). This
means that A lies on the straight line M N since
2
2
I A01 12 - I A0 2 12 = 101M 1 - I 0tM 1 , where
0 1 and O2 are the centres of the circles.
280. Consider the general case of arbitrary circles. Let the points F and F' be arranged as in
Fig. 50. The notations are clear from the figure.
Prove that there is a circle inscribed in the quadrilateral A KBM, and then use the result of Problem 55 of Sec. 2. To this end, it suffices to prove
I

330


that (see Problems 241 and 242 of Sec. 2)

I BF I + I BF' I == I AF' I + I AF I.

(1)

Bearing in mind that I BL I == I BT I, and I FS I =
I FT I, we get: I BF I == I BL I - I FS I, and
similarly, I FA I == I FQ I - I AE I, f BF' I =
I F' P I - I BL I, I F'A I = I A E I - I F'RI. Substituting these expressions into (1), we get:

H
Fig. 50

lBLI-IFSI + IF'PI -IBLI = lAE1-IF'RI +
I FQ 1- IA E I ~ IF'R I + IF'PI = I FQ 1+ 1 S I ~
F
I PR I == I soI. The remaining cases of arrangement

of the points F and F' on the tangents are considered exactly in the same way (the results of
Problems 241 and 242 of Sec. 2 being taken into
account). Since each tangent is divided into four
parts by the points of tangency and the point of
intersection, we have 1/2 X 42 == 8 cases.
To prove the second part of the problem, we
note that the midpoints of A B, F F' and the centre
of the third circle 03' inscribed in A KBM, lie on
a straight line (see Problem 243 in Sec. 2). But
since the radii of the given circles are equal, A B
is parallel to °102 (0 1, O 2 the centres of the given
circles): A and B lie on the straight lines 0t.0:t and


331

°
0 1 8' respectively. Hence, the straight line passing through 0 3 and the midpoint of AB bisects
OlD'.
281. Let M be the point of intersection of the
tangents II' ml' and nl' N the point of intersection
of 12 and m2 (Fig. 51). Through N, we draw a
straight line n2, touching a, distinct from 12 • In
the same way, as it was done in Problem 280 of
/

m,

Fig. 51
Sec. 2, we can prove that the lines mit nit m2' and
ni touch the same circle, this circle being escribed
with respect with the triangle P MQ (it touches
the side PQ), that is, coincides with y. Remark. Figure 51 corresponds to the general case of the arrangement of the circles satisfying the conditions
of the problem.
282. Prove that the straight line DIC passes
through 0, the centre of the arc A B, and the

332


straight line DC I through 0t, the centre of the
arc AB} (Fig. 52). DAD} is a regular triangle,
I DC I = lAC J, consequently, DIG isperpendicular to DA, and DIC passes through o. Analogously,

c,

Fig. 52
DC 1 is perpendicular to D 1A . The point 0. lies on
the arc A B since it is obtained from 0 by rotating
the latter about the point A through an angle of
:r13. Let both arcs be measured by the quantity
6a (for convenience, a > n/6). Then, L A 0t C1 =
2a, L 0IC.A = nl2 - (x, L FACt = 2a. Conse-

quently, L.AFC 1 = n - 2a- ( ~ ~ a) = ~
that is, IAFI = IACtl = IACI.
Let us prove that the triangles FAC and EDC
are congruent. We have: IAFI = IACI = IDCI =

(X=:;LFCtA,
IDEI,

LCDE

=

LCDB -

LBDE

==

It -

2a-

~ ) =2a- ; =
IFCI = ICEI. Further, we find

(n-2 L.DBE)= -2a+2 (2aL.FAC;

L DCE

thus,
2n

==T-a,

n

LB.FD= T-a (measured by

half the sum of the corresponding arcs), L. BtFC =
It
5
n- LCFA= 3+ a , t.DFC==r; n, LDCF =1f,-

333


}n- ~+ ~ = ~ -~

and,

finally,

LFCE=

(--}-~)-(~ -~)=~.
283. Consider two cases: (1) the triangle ABC
i. circumscribed about the given circle; (2) the
given circle touches the extensions of the sides AB
and AC.
In the first case, we consider the circle touching
both the sides of the angle at points M, N and the
circle circumscribed about the triangle ABC internally. Let a, b, c be the sides of the triangle ABC,
r the radius of the given circle, LA = a, I A M I =
I AN I = s: Let us make use of Ptolemy's generalized theorem (Problem 239 in Sec. 2): za =
(b -

z) c

+ (c -

z) b, whence x

=

a

+2bc + c
b

4S ABC
2r
th t .
·
t t
(a+b+c)sina -sina'
a IS,Z18consan.
(It is possible to prove that M N passes through the
centre of the given circle.)
In the second case, we have to take the circle
touching externally both the sides of the angle
and the circle circumscribed about the triangle
ABC.
284. Denote the sides of the triangle ABC in a
usual way: a, b, c; let I BD I == d, I AD I = bl ,
I AM I = a . Use Ptolemy's generalized theorem
(Problem 239 in Sec. 2): xa
(d - bi
z) b =
(b - x) c whence

+

+

(1)

Take on A B a point N such that M N is parallel to
BD. We have:

IMNI

=

:1

d,

IANI =

:1

c,

334
SAMN


=(

:1 r

SABV= (

:1 ) b;
2

SABC

xt

= bIb SABC.

(2)

Let r be the radius of the circle touching M Nand
the extensions of AN and AM. Then from (1) and
(2) it follows that
2X 2SABC

2SAMN

r

= I AM I + I AN 1- 1 N I
M

bx (b 1

+ c- d)
2SABC

a+b+c '
that is, r is equal to the radius of the circle inscribed in thetriangle:ABC, which was to be proved.
285. Let M and K denote the points of tangency
of the circles, with centres at 0 1 and 02' and A C,
respectively. It follows from the result of the pre-

ceding problem that

L.01DM

=

L.OKD =

;

,

L.0 2DK = L.OMD = 90°- ~
We extend OK
and O};[ to intersect 0IM and 02K at points L
and P, respectively (Fig. 53). In the trapezoid
LMKP with bases LM and PK we have:

IMDt

"~1i,' =

IP0 2 1

IDKI = I02KI· Consequently, 0 102 passes
through the intersection point of the diagonals of
the trapezoid-the point 0. In addition,

ILMI
IPKI

IMKI tany
IMKI cot ;

tan! ~

2 ·

286. The statement of the problem follows Irom
the results of Problems 285 and 232 Sec. 2.


335

287. The statement of this problem can be
proved wi th the aid of the result of Problem 240
more precisely, of its particular case, when th~
three circles have a zero radius, that is, they are

H

n

If

Fig. 53
points. In this case, these points are the midpoints
of the sides of the triangle.
288. The statement of this problem follows
from Feuerbach's theorem (see Problem 287 in
Sec. 2) and from the fact that the triangles ABC,
AHB, BRC, and eRA have the same nine-point
circle (the proof is left to the reader).
289. Let in the triangle ABC, for definiteness,
a ~ b ~ c. Denote the midpoints of the sides Be,
CA, and A B by A It B l' and Cit respectively, and
the points of tangency of the inscribed and escribed
circles and the nine-point circle of the triangle
ABC by F, Fa' F b , Fe' respectively. We have to
prove that in the hexagon CtFcFAtFaFb (the
points taken in the indicated order form a hexagon
since a ~ b ~ c) the diagonals CtA It FcFat and
FF a meet at a point. To this end, it suffices to

336


prove (see Problem 49 of Sec. 2) that

I c,r, 1·1 FAt I I FaFb I = I FeF I
X I AtFa I I FbCt I.

(1)

Using the formulas obtained in Problem 201 of
Sec. 1, we find
b-a • /

IFAII

c-b .. I'"
= -2- V

lFaFbl =
IF FI _
c

-

R

V

lelFel = - 2 -

R+2rc'
R

R-2r'

(a+b) R

YR+2r a·YR+ 2rb

,

(b-a) R

VR-2r.YR+2r c '
c-b .. /'

IAtFal = - 2 -

V

a+b .. /
IFbC11 = - 2 - V

R

R+2ra'
R
R+2rb·

Now, the equality (1) can be readily checked.
Remark. I t is possible to prove that the intersection points of the opposite sides of the quadrilateral
whose vertices are the points of tangency of the
inscribed and escribed circles of the given triangle
with its nine-point circle lie on the extensions of
the midlines of this triangle.
290. Using the formulas of Problems 193, 194,
and 289 in Sec. 2 (in the last problem t see its
.
. I FbFcl_ (a+b) (b+c) (c+a) R3
solution), we find. IBIGII --abc.IOlal'IOIbl.IOlci.
The ratios of the other corresponding sides of the
triangles FaFbFc and AIB1C I are the same. The
similarity of the other pairs of triangles is proved
in similar fashion. For I AlB 2 I and the other

337


quantities, we derive formulas similar to that of
Problem f 94, Sec. 2.
29t. Prove that ~ABP == b.ACQ. For this
purpose, it suffices to prove that 6.KBP ==
~ABC and 6.FCQ == LABC (by two sides and
the angle between them): L QA P == L. CA B +
,-CAQ
L.BAP == LCAB
L.CAQ+L.CQA=
L CAB+fBOo- L. QCA = L. CAB-t·900- LQCF =
90° (it was assumed that L CAB ~ 90°; the case
L. CA B > 90° is considered in a similar way).
292. Since L FE}E = L. FCE == 900 , FE}EC is
an inscribed quadrilateral, L FeEl == L. FEEl ==
60°. Analogously, FElAD is an inscribed quadrilateral, and L. E1DF == L. E1AF == 60°, that is,
DEtC is an equilateral triangle. In similar fashion, we prove that BF.C is also an equilateral triangle.
293. Let P, Q, and R denote the points of intersection of LB and ACt AN and BC, LB and
AN, respectively. Let I BC I == a, I AC I === b.
It suffices to show that S ACQ == S APB (both of these
areas differ from the areas under consideration
by the area of the triangle APR). By the similarity
of the corresponding triangles we get ICQI =
ab
1
IPCI = a+b. Consequently, SACQ==21ACr X

+

+

ICQI

ab2

1

== 2(a+b) , SAPB==SACB-SPCB=T ab -

a2 b
2 (a+b)

ab 2
2 (a+b) •

295. Prove that the area of the triangle with
vertices at the centres of the squares constructed
on the sides of the given triangle and located
outside it and the area of the triangle with vertices
at the centres of the squares constructed on the
same sides inside the given triangle are respectively
equal to
22-01557

S

+~

(a 2

+b +c
2

2

)

and

I

S-

~

X

338

Problems in plane Geometry

+b +c

(a2

2

2)

I,

where a, b, and c are the sides

and S the area of the given triangle.
296. Denote: L.A1BC==a, LAICB=~; then
S
AA 1 divides BC in the ratio equal to SABA!
ACAI

1 IABI·IBAll sin (LB+a)
2

~

IACI.ICA11

sin(LC+~)

c

·=T

sin

~

sina X

sin (LB+a)
H·
. d
··1
sin (LC+~).
avmg carrie out simi ar computations for the other sides of the triangle ABC,
use Cava's theorem (Problem 44 of Sec. 2).
297. Let KL be the arc contained inside the
triangle ABC. Extending the sides AB and BC
A,

A

Fig. 54
beyond the point B, we get the arc M N symmetric
to the arc KL with respect to the diameter parallel to A C. Since LB is measured by the arc equal
1
to T(-KL
-MN) = -KL, the arc KL has

+

339


a constant length, and a central angle equal to
the angle B corresponds to it.
298. Let
be the intersection point of the
straight lines, A and Al two positions of the point
on one line of different instants, Band B I the
positions of the other point on the other line at
the same instants (Fig. 54). Erect perpendiculars
at the midpoints of AB and AIB I and denote the
point of their intersection by M: .6.AA 1M =
~BBIM since they have three equal sides: one is
obtained from the other by rotation through the
angle AOB with centre at M. This rotation makes
a point on A go into the corresponding position
of a point on OB so that the point M possesses the
required property.
299. (a) Let A and B denote the points of intersection of the circles, A the starting point of the
cyclists, M and N the positions of the cyclists at a
certain instant of time. If M and N are on the
same side of AR, then LABM = LABN, if they
are on both sides, then LABM
LABN = 180°,
that is, the points B, M, and N lie on a straight
tine. If Land K are two points of the circles diametrically opposite to B (L and K are fixed), then,
since LLNM == LNM K == 90°, the point P which
is the midpoint of LK is equidistant from Nand
M. We can make sure that P is symmetric to the
point B with respect to the midpoint joining the
centres of the circles (Fig. 55, a).
(b) Let 0 1 and 0t denote the centres of the
circles. Take a point A 1 such that 0IA 0IA 1 is a
parallelogram. It can be easil y seen that the triangle
MOlAl is congruent to the triangle NOtAl. since
IMOII = lOlA I = I 0sAl I, IOIAll = IOtA 1=
INO,I, LM01A 1 ~ q>+LAOIA l =q>+LAOjA t =
LNO,.A 1 , where q> is the angle corresponding
to the arcs covered by the cyclist (Fig. 55, b).
Thus, the sought-for points are symmetric to the
points of intersection of the circles with respect
to the midpoint of the line segment 010t.
Remark. In Item (a) we could proceed just

°

°

+

22*

340


34t

in the same way as in Item (b). Namely, taking
the point P so that 60 tP0 2 = 60 tA0 2 (A and
P are on the same side of 1° 2 and do not coincide),
it is easy to prove that the corresponding triangles
are congruent.
300. (b) Use the result of Item (a). Replace the
rotation about 0 1 by two axial symmetry mappings,
taking the straight line 0 101 as the axis of symmetry for the second mapping and the rotation
about the point O 2 by two symmetry mappings,
taking the straight line
2 as the axis of the
symmetry for the first mapping. Remark. If ex
~ = 2n, then the application of the given rotations
in succession, as it is easy to make sure, is
equivalent to a translation.
A nsuier: if a
p < 211:, then the angles are
a
P
a+p ,and If ex p>
.
equal to 2' 2' n 2

°

0.°

+

+

+

221,

then the angles are equal to

21 -

~

,21-

~

,

a-l- P

-2301. Let us carry out three successive rotations in the same direction about the points K, L,
and M (or about Kit L 1 , and M t ) through the
angles a, p, and y. Since a
p y = 2n, the
transformation obtained in a translation (see Problem 300 in Sec. 2). But since one of the vertices
of the original triangle remains fixed in these
rotations, all the points of the plane must remain
fixed.
Thus, the centre of the third rotation (the point
M) must coincide with the centre of the rotation
resulting from application in succession of the
first two rotations: about the points K and L.
Now, take advantage of the result of the preceding
problem.
LDOE = 2p,
302. Denote: LBOC = 2a,
LFOA = 2y. Let K, M, and L be, respectively,
the intersection points of the circles circumscribed

+ +

342


about the triangles BOC and AOF, BOC and
DOE, AOF and DOE. The point K lies inside the
triangle AOB, and LBKO == 180° - LBCO ==
90° + a, LAKO = 90°
y, and since a
13+
y = 90°, LAKB = 90° +~. Similarly, L lies
inside the triangle FOE, and LOLF= 90°
y,
LOLE=90o+~, L FLE=900+a. Hence, lOLl ==
lAKl, LKOL = 2y + LKOA + LLOF = 2,,+
L. KDA + L. KAO = 90°+" = LAKO; thus, the
triangles KOL and AKO are congruent, that is,
I KL I == lAO I=R. We then prove in a similar

+

+
+

way that each of the two other sides of the
triangle KLM is equal to R.
303. Let ABCD denote the given quadrilateral,
0h 02' o; 0 .. the centres of the rhombi constructed
on AB, Be, CD, DA, respectively; K and L the
midpoints of the sides A Band BC, respectively,
M the midpoint of the diagonal A C. The trian-

gles 0lKM and 0ILM are congruent

~ I AB I

I LM I,

I KM I

101LI. L01KM=LO ILM)

(I 0lK I=

~ I BC I =

If LABC+<X<

n, then these triangles are located inside the trian-

+

gle OlMOs, and if LABC
a > 11, then they are
found outside the triangle 0lM0 2 (the angles of
the rhombi with vertex at B are equal to a). Thus,
I 0iM I == 10 1 M I, LOlM0 2 = 11 - a. In similar fashion, I DaM I = 104 M I, LOaMO, ==
1t a. Consequently, the triangles 0IM0 3 and
02MO.. are congruent, and one is obtained from the
other by a rotation about M through the angle
1t a. Hence, there follows the statement of the
problem.
304. Let ABC be the given triangle, A tBtCt
the triangle ~, A 2B 2C, the triangle 6 (At and
A 2 the centres of the triangles constructed on Be),
a, b, c the sides of the triangle A Be.
(a) The fact that AIBiCl and A 28 I C I are reg-


343

ular triangles follows, for instance, from the result of Problem 301, Sec. 2.
(b) Let us prove a more general assertion. If
on the sides of the triangle A BC there constructed
externally (or internally) similar triangles A tBC,
BICA, CIAB so that LAIBC = LB1CA ==
LelAB, LAtCB == LBIAC = LCIBA, then the
median points of the triangles ABC and AtBIC.
coincide. First note that jf M is the point of intersection of the medians of the triangle ABC, then
~

+

MA

~

MB

+

~

MC == 0, and, conversely, if this

equality is fulfilled, then M is the median point of
~

+
MB + MC = 0
or (MA + ACt) + (MB +
-+
-+-+BA + (Me + CB t ) = o. But MA + MB +
-+
-+
-+
MC = O. In addition, ACt + BA + CB = 0
-+ -+ --+the triangle ABC. It remains to check that MAl
-+-

~

t

~

-+

-+-

I

~-+-

I)

~

I

I

since each of the vectors A CIt BA t, CRt is obtained
-+

-+

~

from the vectors AB, BC, CA, respectively, by
rotating the latter through the same angle (LAtBC)
and multiplying by the same number.
(c) Consider a more general case. The isosceles
triangles AIBC, BICA, CIBA and A;BC, BiCA,
C ;BA in which the ratio of the length of the alti tude drawn to the base to the length of the base
is equal to k are constructed on the sides of the
triangle A Be externally and internally as on
bases. Let 0 denote the centre of the circle circumscribed about the triangle ABC; a, b, c its sides;
A 0' B o' Co the midpoints of BC,CA, A B, respectively. For definiteness, we assume ABC to be acute

triangle. Then,

~

(IOAol

SAIOCl=

~ IAIOI·leIOI sinB =

+ ka) (IOCol + kc) sin B

=

~

IOAol X

344


~

IOCol sinB+

klaesinB+

~ (aIOCol+eIOAol)x

sin B = k 2S ABC + S AoOCo+ : i». Obtaining similar
relationships for the triangles AIOB I and BIOC I
and adding them together, we find: S AIBICI =

1)

( 3k l +
S ABC + : (a l + bl + el ) (this equality
is also valid for an obtuse triangle ABC). For the
k
triangle A~B;C; we have: S A'B'C' == 4 (a 2 + b2
el )

(al

-

I·

+

+ 1)S ABC Consequently if
+ b + e (3k + 1)S ABC ~ 0,
(

3k l
l

SAIBICI (a l

I

III

+

bl

l

l

) -

l

) -

+ 1)SABC < 0,

k

SAIBICI- S A'B'C' = -2 ·(a2
a2

+ +
bl

that always
lem 362 of Sec. 2,

then

l

(3k l

1 1 I

: X

(6k +nSABc,andif~X

SAiBiCi =

+e

I

+b +c
2

2

).

then

We can prove

;a, 4 ya S ABC (in Proba stronger inequality is

c2

proved), and this means that for k =

~

the

2V3

difference between the areas of the triangles
AIBtCI and AiB~C; is equal to SABe.

305. Let the three given points form a triangle
ABC. Two families of regular triangles circumscribed about the triangle ABC are possible. The
first family is obtained in the following way.
Let us construct circles on the sides of the triangle
so that the arcs of these circles lying outside the
triangle are measured by the angle of 4n/3. We
take an arbitrary point A I on the circle constructed
on BC. The straight line A IB intersects the circle
constructed on BA for the second time at a point


345

Cit and the straight line AlC intersects the circle
constructed on CA at a point B I . The triangle
A IB1CI is one of the triangles belonging to the
first family. Let E, F, and G denote the intersection
points of the angle bisectors of the triangle AtBtCl
and the circles constructed on the sides 01 the
given triangle. The points E, F, and G are fixed
(E the midpoint of the arc of the circle constructed
on BC and situated on the same side of BC with the
triangle ABC). The points E, F, and G are the
centres of the equilateral triangles constructed
on the sides of the triangle A BC internally. The
triangle E FG is a regular one (see Problem 304 in
Sec. 2), its centre coinciding with the median point
of the triangle ABC. The centre of the triangle
AIBICt lies on the circle circumscribed about the
triangle E FG; the square of the radius of this
2
2
2
1.
h
circle being equal to 9' ( a
2 c
28 V 3) ,

+ +

where a, b, and c are the sides and 8 the area of
the triangle A BC (see the solution of Problem 304, of Sec. 2).
The second family of equilateral triangles circumscribed about the triangle A BC is obtained if
on the sides of the triangle ABC circles are constructed whose arcs located outside the triangle
ABC are equal to 2n/3 (each).
The required locus consists of two concentric
circles whose centres coincide with the median
point of the triangle ABC, and the radii are equal

to

t .. /" 1

"3 V 2

(a2

+

h2

+ c ± 28 V~3.
2

)

306. Prove that the triangles CB tA 2 and CAtB!
are obtained one from the other by rotation about
the point C through an angle of 90°. Indeed,
~CBBI (I BB i I = I AC I,
I BC I =
IAA11, LCBB t = LCAA t ) , and since AA t 1- BC
and BB t 1- AC, we have: BtC 1. AtC. Similarly,
~CAAl =

AtC and BIC are equal to each other and mutually
perpendicu ar.

346

Problems in Plane G·eometry

307. Prove that the tangents to the circle drawn
from the vertices between which one of the vertices
of the polygon is located are equal to each other.
Hence, it follows that for a polygon with an odd
number of sides the points of tangency are the
midpoints of the sides.
308. Note that if we consider the system of
vectors whose initial points lie at the centre of
the regular n-gon and whose terminal points are
at its vertices, then the sum of these vectors equals
zero. Indeed, if all of these vectors are rotated
through an angle of 2n/n, then their sum remains
unchanged, and on the other hand, the vector equal
to their sum rotates through the same angle.
Hence, the sum of the projections of these vectors
on any axis is also equal to zero.
Let us return to our problem. If q> is the angle
between the given straight line (let us denote it
by l) and one of the vectors, then the remaining
2n
2n
vectors form the angles cp+-- , q>
2- ,
n
n
q>
(n - 1) 2n . The square of the distance from
n

+

+

+

the kth vertex to l is equal to sin! (cp k 2:

~

(1- cos ( 2cp

+k

+

:n ))

Y
=

But the quantities

( 2cp k 4: ) can be regarded as projections on
l of the system of n vectors forming angles 2q>
k 411 (k=O, 1,
n-1) with t, If It is odd,
n
these vectors form a regular n-gon, if n is even,
then they yield an ; -gOD repeated twice.
Answer:

+

n

2.

309. (a) If the side of the polygon is equal to
%:U • • • , %n are distances from

at S is its area, %1'

347

a certain point inside the polygon to its sides,
then the statement of the problem follows from the
aXg
ax n }/2.
equality S = (axJ
(b) Consider the regular polygon containing
the given one whose sides are parallel to the sides
of the given polygon. The sum of distances from
an arbitrary point inside the given polygon to the
'sides of the regular polygon is constant (Item (a»
and differs from the sum of the distances to the
sides of the given polygon by a constant.
310. Let Bit B 2 ,
., B n + l denote the points
symmetric to At, A 2' ••• , A n+l with respect to
the diameter A oA2n+I' C k and Ci the points of intersection of the straight line A kAsn+l-k with OA n
and OA n+l. Let D k - l and D k be the points of intersection of the straight lines A hBk-1 and A kBk+l
with the diameter. Obviously, the same points are
the points of intersection of the straight lines
BkA k- 1 and BkA k +1 with the diameter. It is also
obvious that the triangle Dk-1AkD k is congruent
to the triangle CkOCk. Thus, the sum of the line
segments eke;" is equal to the sum of the line segments Dk-1D k (k = it . · .t n), Do = A o' D n =
0, that is, equals the radius.
311. Let A (Fig. 56) be the given point, A k a
vertex of the 2n-gon, B k-l and B k the feet of the
perpendiculars dropped from the. point A on the
sides enclosing A k» and elk and ~k the angles
formed by the straight line AA k with those sides
<Pit = LAAkB k- 1t ak = LAAkB k). Since a circle
can be circumscribed about the quadrilateral
ABk-1AkB k,
we
have:
LABA-1B k = ak'
LA BkB k- 1 = Pk (or supplement these angles to
180°)· thus by the law of sines, IABA-li

+

t

,

+

.+

IABkl

sin Ph
IABk-11 IABk+11
sin ~k sin ak+l
sin ak '
IABk 12
sin ak sin ~k+l '.
Multiplying those equalities for k = 2, 4, ... ,

348


+

2n and replacing the index 2n
t by t, we
get the desired result.
312. Prove that if Ok and 0k+1 are the centres
of the circles touching the given circle at points
A", and Ak+l; B the point of their intersection lying
on the chord AkA",+!; rk' rk+1 their radii,
rk +1 = r, LAkOkB = LA k +1 0 k +lB =
then rk
LA ADAk +1 (r the radius of the given eire e, 0

+

A

Fig. 56
its centre). Hence it follows the equality of every
other radii, which for an odd n means that all of
them are equal to r/2. In addition, -A kB
'-'BA It +1 = -- A h,A It +1 (the minor arcs of the corresponding circles are taken).
313. (a) Let A be an arbitrary point of the
circle (A on the arc A tA tn+1). Let a denote the
side of the polygon, and b the length of the diagonal
joining every other vertex. By Ptolemy's theorem
(Problem 237 in Sec. 2), for the quadrilateral
AAAA k +1A k +1 we have: I AA k I a+ I AA k +21 a=
IAAk+ll b
(k = t , 2, .. , 2n - 1).
Similar
relationships can be written for the quadrilaterals A 2n A 2n+1AAI and A 2n+l A A t A 2:

+

I AA 1 I a + I AA 2n+1 J b == , AA 2n I a,
I AA In+l I a + I AA 1 I b = I AA I I a.

349


Adding together all these equalities and leaving
even vertices on the right and odd vertices on the
left, we get the required statement.
(b) Our statement follows from Item (a) and
the result of the Problem 206 of Sec. 1 (A similar
formula can be obtained for the case of internal
tangency.)
314. (8) Let l intersect A C and BC at points
K and N, respectively, and touch the circle at
a point M (Fig. 57). Let us denote: I A C I ==

c
Fig. 57

= x, I BN I ==
2
w
(a-x) (a-y)
'uv
xy
but, by the law of cosines, for the triangle CK N
the following equality holds true: (x
y)2 ==
(a - X)2 + (a - y)2 - 2 (a - x) (a - y) cos a =>
. 2 a
xy
Th
uu _ . 2 ex.
sln 2
(a _ x) (a _ y)
us, 7 - SIn "2.
(Other cases of arrangement of the line I are considered in a similar way.)
(b) Let us use the result of Item (a). Multiplying
the corresponding equalities for all the angles of
the n-gon, we get the square of the sought-for ratio, and the ratio itself turns out to be equal to
I Be I = a,

I A K I = I KM I

I NM 1=

Obviously

y.

+

1/ (sin ~' sin ~z
sin ~n )
a., ..., an arc the angles of the

where
polygon.

!Xl.

350


(e) We use the result of Item (a). We denote
the points of tangency of the sides AlA. 2' A 2 A 3'
A2n-IA2n, A 2nA I with the circle by Bit B 2 , - _,
B s n - I , B 2n t respectively; the distances from
At, A 2' - - -, A In to l by Zit X 2, - -, X2n-l, %In,
respectively; the distances from B 19 B 2 ,
Bin
to l by Yl' Ys,
Yin' respectively Then we
get:
t
x~
1
0

-

0'

0

sin2~
2

Y2nYI

--=L==
YIY2

sin 2 !:!
2
1

Y2n-1Y2n

sin2 al n
2
where (Xl' a
0' a 2 n are the angles of the
polygon. Mu tiplying the equalities containing
Xl' Xs, • - 0' X2n-l and dividing them by the product of the remaining equalities, we get:

, ·.
f

%IX3 - _. Xln-l
(

X,X4

) 2

x2n

.

a2

.

a4

Sln-Sln-

2

(

2

.
aJ
• (xs
SlDTS1DT

sin-nex2

a.t~-l

)2

S l n - -o

2

315. The statement of the problem can be proved by induction. The beginning step of the proof,
n = 4, is considered in Problem 235 of Sec. 2.
However, we can suggest another way of solution based on the following equality _ Let in the
triangle A Be the angle A be the greatest, rand R
the radii of the inscribed and circumscribed circles, respectively, da, db, and de the distances from
the centre of the circumscribed circle to the corresponding sides of the trtangle. Then

r+ R

= dCJ

+ db + de

(1)


35!

for an acute triangle and
r
R == - da
db
de

(2)

+

+ +

for an obtuse one (for a right triangle, d a = 0 and
for it any of the above relationships holds true).
Proof. Let ABC be an acute triangle; A o' B o'
Co the midpoints of the sides BC, CA, A B, respectively; 0 the centre of the circumscribed circle.
By Ptolemy's theorem (Problem 237 in Sec. 2),
for the quadrilateral ABoOC o we have:

~

dc

+

W··
. °1
riting two more SImI ar re Ia to
ionships for the quadrilaterals BCoOA o and CBoOA o
and adding them together, we get:
c b==T
2 daR

°

(; + ~ ) dc + ( ; + ~ ) db+ ( ~ + ~ ) d a
1

=2 (a+b+c) R=pR,
1
+ db + de) - '2 (cdc + bd b + ada) =
~ (cdc + bdb + ada) = S = pr, after

whence p (d a
pR.

Since

reducing by p, we get the equality (1). The case
LA > 90° is considered in a similar way.
The statement of the problem follows from
the relationships (1) and (2). To this end, let us
write the corresponding equalities for all the triangles of the partition. Note that each of the diagonals serves as a side for the two triangles. Consequently, the distance to the chosen diagonal enters
the relationships, corresponding to these triangles,
with opposite signs. Hence, adding together all
these equalities, we get (provided that the centre
of the circle lies inside the polygon): ~ r R =
d 1+d2
+d n , where d t , d 2 , ••• , d n are the
distances from the centre of the circle to the

+ ...

+


352

sides of the polygon. If the centre of the circle
is outside the polygon, then the distance to the
greatest side should be taken with the minus sign.
316. Consider, for definiteness, the case when
the point M is found inside the polygon. Let u
and v denote the distances from M to AlA 2 and
AlA n' respectively; x and y the projections of
AtM on A IA 2 and AtA (x and y should be assumed to be positive, if these projections are situated on the rays A 1A a and AlAn, and negative
otherwise). 1 AtB I I = IAtB n 1 = a, LAtA1A n =
a. The distances u and v can be expressed in terms
of x and y: u = _.y_-x c~ a
vc::---:-SIn a
SI n a '
SIn a
cos a
t - cos a,
y sin a ; hence u
v = (x
y)
sin a
-

+

(x

+

y) tan

a

2 =

(x

r
+ Y) (;e

2
2
(I MB t 1 + I MB n 1 )

+ (y -

= «x -

+
+

a)2
a)2

v2) a
(u -

+

a

r)2

+ 2r (u + v) - 2r a
=== 2d a + 2ra (u + v) 2

2

=

We

«x -

now have:

+u

a)1

+ (y -

a)2

2

+ (v -

r)2

)

2r2a === 2d2a

+ 2r

2

(x

+ y) -

2r2a.

Writing similar equalities for each of the vertices
and adding them together, we get the statement of
the problem.
317. Consider three triangles ABC, ACD,
and ADB having a common vertex A. Denote the
projections of M on AB, AC, and AD by B t , Ct ,
and D., respectively. The straight lines BIC t ,
CtD h and DtB t are Simson's lines of the point M
with respect to the triangles ABC, ACD, and
ADB. But the points A, M, B t , CI , and D I lie
on the same circle (A M being its diameter). Consequently, the projections of the point M on B I Cit


353

CtD 1, and D1B. lie on a straight line which is

the Simson line of the point M with respect to
the triangle BICtD I. Considering then the projections of the point on Simson's lines corresponding to the three triangles with a common vertex
B, we get that those three projections also lie on
a straight line, hence, the four projections are collinear.
The passage by induction from n to n + 1 is
performed exactly in the same way.
318. Let, for definiteness, B. lie on the arc AlA 2
which bounds the segment not containing the circle

Fig. 58
Let CI , c;
denote the points of tangency
of AlA 2' A sAa,
with the circle p, respectively;
D I , D s'
• the points of tangency of BIB'}.,
BIBs,
with the same circle (Fig. 58); K, L,

Ji.

23-01557

354


and P the points of tangency of D.CI and AIB I,
DICt and AgB I , AIB I and A 2B g •
In the triangles A1KC I and D ILB 2 , we have:
LKCIA I = LLD IB 2 , LC.A 1 K = LD 1B 2L; hence,
LC1KA I = LD I L B 2 , that is, KPL is an iso-

sceles triangle, I K P I = I P L I.
Consider the circle y touching K P and PLat
points K and L, respectively. The centre of this
circle is found on the straight line passing through
the centres of ex and ~ (see Problem 12 in Sec. 2).
Let the line D 2C'}, intersect A 2B, and AaB a at
points L' and M, respectively. As In the preceding case, let us prove that there is a circle y' with
centre on the straight line passing through the
centres of ex and p and touching A 2B 2 and A 3 B 3
at points L' and M, respectively. Let us prove that
y and V' coincide. To this end, it suffices to prove
the coincidence of Land L' We have:

:~~~:

s A2CIDI

21IDICll·IAsCtlsinL.AIClDl

SB2 C I D l

~

t:

ID1C11·IBIID11 sin LBlID1C 1

S· °1 I IAzL'1
IAzCII
IAICII
nm ar y IL'B11
I BID I I
: IBi D 1 1 '
that is, Land L' coincide. Remark. It follows from
our reasoning that in the case under consideration
the points of tangency of y with the straight lines
A 1B lt A IB 2 , •• , are fOfnd inside the line segments AlB I , A tB 2 , •• ~
319. Using the notati n of the preceding problem, the statement is re uced to the following: if
An +1 coincides with
1, then B n +1 coincides
with B I • Suppose. the contrary. Then AIB I and
AlB n +1 touch the circle "I, A l A 2 intersects y, and
B I and B n +1 lie on the arc AlA 2 corresponding to
the segment not containing ~. The points of tangency of AIB I and AIB n + t with y lie inside the
line segments AIB I and AlB n +l . Thus, we have
obtained that two tangents are drawn from Al to
IAICII

IB2 D 1 1

•

355


"It the points of their contact with y being located
on the same side of the secant AlA 2. But this is impossible.
320. Let us consider the triangle BoXC o. The
straight line X R is the bisector of the angle CoX B o'
It is readily checked that LCoRB o = ~

+

~ LCoXB o- Hence, it follows that CoR and BoR
are the bisectors of the angles XCoB o and XBoC o,
respectively (see Problem 46 in Sec. i). In similar
fashion, in the triangles CoYA 0 and A oZB 0 the
points P and Q are the points of intersection of
the angle bisectors. Hence, taking into consideration that LPAoQ = LA/3, LQBoR = LBI3,
LRCoP = LC/3, we get the statement from which
Morley's theorem follows.
32t. When solving the problem, we use the
followinf assertions which can be easily proved.
(a) I a point N is taken on the bisector of
the angle M of the triangle KLM (inside this
triangle) so that LKNL = ~ (31
LKML),
then N is the intersection point of the angle bisectors of the triangle KLM (see Problem 46 of
Sec. t).
(b) If a point N is taken inside the angle KM L
and outside the triangle KLM on the extension
of the bisector of the interior angle M so that

+

LKNL

= -}

(31 -

LKML), then N is the in-

tersection point of the bisector of the angle M and
the bisectors of the exterior angles K and L.
(c) If a point N is taken inside the angle KM L
and on the bisector of the exterior angle K of the

triangle KML so that LMN L = ~ LMKL,
then N is the intersection point of the bisector
of the angle M and the bisectors of the exterior
angles K and L.
23*

356


We carry out the proof of the assertion for all
possible values of t, t, k (all in all, seven cases)
according to one scheme. Each time we formulate
and prove the corresponding converse assertion
equivalent to the considered case of Morley's theorem. The preceding problem is an example of
following such a scheme. In order to avoid repetition, let us first single out the general part of reasoning. Consider the regular triangle PQR. Constructed on its sides as bases are isosceles triangles
PXQ, QYR, RZP (what triangles and how they
are constructed is explained for each of the seven
cases). Let A 0 denote the point of intersection of
the straight lines ZP and YQ, Bothe point of
intersection of XQ and ZR, and Co the point of
intersection of YR and X P. Then we prove for each
case that the triangle AoBoC o is similar to the
triangle ABC, and that the rays AoP and AoQ,
BoQ and BoR, CoR and CoP are its angle trisectors
of the corresponding kind.
Let us now indicate what triangles and how they
should be constructed on the sides of the triangle
PQR in each case.

=j=

=

k

1; LPXQ

=

f

+

(31
2 LA),
1
LQYR= 3 (n+2 LB), LRZP=3 (n+2 LC).
All the triangles are arranged externally with
respect to the triangle PQR.

(1) i

1

~ (31-2 LA),
2LB
2LC
LQYR = l' - - - LRZP = n - - 3
'
3
·
All the triangles are arranged externally with
respect to the triangle PQR. (We assume that
LA < n/2. If LA > n/2, then the triangle P XQ
is "turned out" on the other side of the triangle
1
PQR, LPXQ = 3" (2 LA - n). If LA = n/2,
(2) i=1, j=k=2, LPXQ=


357

then the triangle P XQ turns to a pair of parallel
straight lines. This note should be borne in mind
when considering the further cases.)
(3) i

=

j=1, k=3; LPXQ=

~

(n-2 LA),

1
t
LQYR=T (n-2 LB), LRZP = 3" (n+2 LC).
The triangles P XQ and QYR are arranged externally and RZP internally with respect to the
triangle PQR {see Item (2».
(4) i = j = k = 2; LPXQ =

~

(n - 2 LA),

t

1

LQYR= 3 (1t-2LB), LRZP=3 (1t-2 LC).
All the triangles and the triangle PQR itself are
arranged on the same side of the corresponding
sides of the triangle PQR, (see Item (2».

+

(5) i=1, j =2, k=3; LPXQ=+ (n
2LA),
1
2LC
LQYR=T(n-2 LB), LRZP=n-The
3triangle P XQ is constructed externally with
respect to the triangle PQR, while the other two
internally (see Item

2».

(6)

i=2,

j=k=3;

LPXQ=n-

2~A

,

1
1
LQYR=3 (n+2LB), L RZP==T(n + 2 LC).
The triangle P XQ is arranged externally and the
two others internally with respect to the triangle
PQR.
(7) i=j=k=3; LPXQ=n2LB
n - --

2~A

, LQYR=

2LC
.
a , LRZP=1t--3- . All the triangles are arranged inside the triangle PQR .
Item (1) was proved in Problem 320, Sec. 2.
Let us, for example, prove Item (2).

358


Let LA < 1t/2. Consider the triangle BoXCo
in which XR "is the bisector of the angle BoXC o.

In addition, LBoRC o = ~ (:It
LBoXCo). In
accordance with the assertion (a), R is the intersection point of the angle bisectors of this triangle
(if A > 'Jt/2, then BoR and CoR are the bisectors
of the exterior angles of the triangle BoXC o). Further, in the triangle CoYA o we have: YP is the
bisector of the exterior angle Y, LAoPC o =
LA YCo (this can be readily checked). In accordance with the assertion (c), P is the intersection point of the bisector of the angle C oA 0 Y and the
bisectors of the exterior angles A oCoY and CoY A 0
of the triangle CoYA e- In similar fashion, the
point Q with respect to the triangle A oZB 0 is the
intersection point of the bisector of the angle ZA oB0
and the bisectors of the exterior angles A oZBo
and A oBoZ. (This implies that the triangle PQR,
with respect to the triangle AoBoC o, is formed by
the intersection of the trisectors of the first kind
of the angle A 0 with the trisectors of the second
kind of the angles B o and Co (Item (2) is meant).)
The triangle A oBoC0 itself is similar to the triangle
ABC.
In all the remaining itemt(from 3 to 7) we reason in a similar way varying only the assertion
used «a), (b), (c».
I nterchanging the indices i, t, k, we note that
to Itern 5 there correspond six regular triangles,
to each of I terns (2), (3), and (6) three regular
triangles, to each of Items (1), (4), and (7) one equilateral triangle. Thus, the total number of regular
triangles obtained is eighteen.
Now, in each case we choose the dimensions of
the triangle PQR so that the corresponding triangle
AoBoC o is equal to the triangle ABC. We superimpose the eighteen obtained drawings by turns so
that the trianglesABC are brought into coincidence.

+

+


359

It should be done in the following succession:
first, we take the drawing corresponding to Itern
(1), then the three drawings corresponding to Itern
(3), then the six drawings corresponding to Item
(5), then the three drawings corresponding to
Item (2), and, finally, the three drawings from Item
(6), one from I tern (4) and one from I tern (7). In
each successive superposition, at least one of the
vertices of the corresponding regular triangle
must coincide with one of the vertices of the triangles already superimposed. If we count the angles
we can see that five vertices of two equilateral
triangles, having a common vertex, lie on two
straight lines passing through this common vertex.
Thus, the vertices of all the eighteen equilateral
triangles "must" be arranged, without fail, as in
Fig. 59. (In this figure, alP; denotes the point of
intersection of the trisectrices at and P;, etc.).
322. For the equilateral triangle with side
equal to 1 the radius of each of Malfatti' s circles
.
~3-1
IS equal to
4
• The sum of the areas of the
·
correspon di ng eire Ies equa1s 3n (2 -V"3)
1
8

. An d

the sum of the three circles one of which is inscribed in this triangle and each of the two others
touches this circle and two of the sides of the
..
iin
3n (2- V3)
triangle IS equal to 108 >
8
•
323. Use the equality Rr=

+ +

~~

and inequal-

ity 2p = a
b
c:>- 3 Vabc (the mean-value
theorem).
324. If PI is the semiperimeter of the triangle
with its vertices at the feet of the altitudes of the
given triangle; p, S, r, and R the semi perimeter,
the area, the radii of the inscribed and circumscribed circles, respectively, then S = pr and, in
addition, S = P1R (the latter follows from the fact

360

that the
into the
the line
dropped

radius of the circumscribed circle drawn
vertex of the triangle is perpendicular to
segment joining the feet of the altitudes
on the sides emanating from this vertex).
r
1
Consequently, Pl=PR~2 p .

Fig. 59
325. Let m a be the greatest of the medians.
If we use the relationship m~ > mi
m~, following from the hypothesis, and replace the medians
by the sides a, b, and c of the triangle (Problem 11
c2 , whence cos A >
of Sec. 1), we get: 5al < b2

+

2

+

2

2 (b
e
5be

)

~
5

+

(-!?"'+.!-) ~ i- > Vi
c
b
~ 5
2

·


361

326. Let 0 denote the intersection point of
the diagonals of the quadrilateral A BCD. Suppose
that all the angles indicated in the hypothesis
are greater than 1t/4. Then, on the line segments
OB and DC, we can take, respectively, points B 1
and Ct such that LBtAO = LOB1C1 = 1t/4. Let
LBOA = a > n/4. We have:

lOCI> IOCII =

IO"Btl

V2sin (a- ~)

IOAI
•

2 sin
In

(

'It )

a. - 4

similar

sin

(

Ct

+T

-~~IOAI.

'It )

fashion, we prove

cos 2a ~

the inequality

I OA I > I DC r. Thus, we have arrived at a

contradiction.
327. Let the sides in the triangle ABC be related by the inequalities c ~ b ~ Q. We take on CB

a point M such that LCAM= ~ LC. Now, we
a
have to prove that I eM I ~ 2 . By the law of
sines, for the triangle CAM we have: I CM I =
. C
b slnT

. 3C
sln-2

328. Let D denote the midpoint of A c. We erect
at D a perpendicular to A C and denote the point
of its intersection with BC by M. AMC is an isosceles triangle, hence, LMA C = LBCA. By hypothesis, A BD is also an isosceles triangle,
LABD = LBDA, LABM > 90° (by hypothesis),
LADM = 90°, hence, LMBD > LMDB, and
IMDI > IBMI. Hence it follows that LMAD >
LMAB (if B is mapped symmetrically with respect to the straight line AM, then we get a point B 1

362


inside the angle MAD since M D is perpendicular
to AD and I MD I > I MB I = I MB I I); thus
LC> LA - LC, LC >{-LA.
329. If the circle touches the extensions of the
sides AB and AC of the triangle ABC, and its
centre is 0, then it is easy to find that LBOC =

900

-

i-

~

Thus,

LA.

LBOC

+ LA

=

900

+

0

LA =i= 180 •
330. Let AD denote the altitude, AL the angle
bisector, AM the median. We extend the angle
bisector to intersect the circle circumscribed about
the triangle at a point A I. Since MAl is parallel
to AD, we have: LMA1A = LLAD.
A nsuier; if LA < 90°, then the angle between
the median and angle bisector is less than the angle
between the angle bisector and altitude. If LA >
90°, then vice versa; if LA = 90°, then the angles
are equal.
33t. If AD is the altitude, A N the median, M

the median point then cot B

ICDI

ICBI

ICBI

+ cot C=
ICBI

2

:~~:

+

IADI ~ IANI =::3IMNI==a·
332. From the fact that SBAM = SBeM,
I BC I > I BA I, and I CM I > I MA I it follows
that sin LBAM> sin LBCM. Hence, if the
angles are acute, then LBAM> LBCM; only
the angle BAM can be obtuse. Thus, we always
have: LBAM > LBCM.
333. If I OA I = a, R the radius of the circle,
K the point of intersection of OA and DE, then
a"-R2
it is easy to find that I OK I = a - ~=

lADT==

a2

+ R"

24

>

n.

363


334. The notation is given in Fig. 60. In the
first case (Fig. 60, a), IABI<IAAII+IAtBII+
IB1BI = IAAII + IAIGI
IBIDI
)BBII =
IA C I + IBD I. In the second case (Fig. 60, b),

+

+

a

Fig. 60

IABI > IBKI - IAKI > I BE I - I AC I. The
converse can be readily proved by contradiction.
335. Let K, L, and M denote the points at
which the drawn lines intersect A C; we further
denote: I AC I = b, I BC I = a, I AB I = c,
I BL I = I. By the theorem on the bisector of an
interior angle. we find: ILCI

=

a+b ; applying
a c
this theorem once more to the triangle BeL, we find
ba
l
ba
a)
ILMI=--·--==-- ( t - - - ; but
a+c l+a
a+c
a+l
LBLA

={- LB + LC =

n-L;+LC

>

LA

(since LC>3 L.A-ll). Hence, c>l and ILMI
ba

<

a)
ac
b
a c 1- a c = b (a e)i ~ T ·
336. Let ABCD be the given quadrilateral.
Consider the quadrilateral ABICD, where B l is
symmetric to B with respect to the midperpendicular to the diagonal AC. Obviously, the areas
ABCD and ABICD are equal to each other, the
sides of the quadrilateral ABICD, in the order of
traverse, are equal to b, a, c~ d. For this quadrilat-

+

(

+

+

364


eral, the inequality S =::;; ~ (ae
bd) is obvious,
the equality occurring if LDAB 1 = LB""CD ==
90°, that is, ABtCD is an inscribed quadrilateral
with two opposite angles of 90° each; hence, the
quadrilateral ABCD is also inscribed (in the same
circle), and its diagonals are mutually perpendicular.
(1) The given triangle (ABC) is acute. Let
LB be the greatest: 60° ~ LB < 90°. Since the bisectors of the angles A and C are less than 1, the
altitudes of these angles hA and he are also less

+

hAke

va

than 1. We have: SABC= 2 sin B <-3-·
(2) If one of the angles of the triangle, say B,
is not acute, then the sides containing this angle
are less than the corresponding angle bisectors,
that is, less than 1, and the area does not exceed 1/2.
338. Let c be the greatest side lying opposite
the vertex C. If a2
b2
c2 - BR2 > 0, then
2 + b2 > 8R2 a
c2 > c2 (since c ~ 2R), that is,
the triangle is acute. Conversely, let the triangle
3
be acute, then a2
b2
c2 = 2m~+2c2 (mr. the

+ +

+ +
+ +

median to the side c); therefore, the less the median, the less the sum a2
b2
c2 • But the median is maximal if C is the midpoint of the arc and
its length decreases as C displaces in the arc. When
the triangle becomes right-angled, the sum a 2
b2
c2 - 8R2 is equal to o.
339. Replacing Rand r by the formulas R ==
abc
S
.
~, T= p' for computing S make use of
Hero's formula and the equality
abe
abc
S)
48 1 ( p - - - - S ) ( p+-.+-28
p
28 . p
~ (at+bt-e t) (at-bt+e t) (-at+bt+e t).

+

=

+

365


340. Let us assume the contrary, for instance,
+ a > b; squaring the inequalities and adding them together, we get:
5e2 > a2
b2 , which is a contradiction.
341. The bisector of the angle B is the bisector
of LOBH, and the bisector of the angle A is the
bisector of L OAB. Further, L BAH = 90°- L B <
900 - L A = L A B H ; hence, IAHI> IBHI. If K
and M are the intersection points of the bisectors
c::>- a; then 2c::>- c

+

of the angles A and B with OB, then

IABI

IABI

IBNI

IBNI

II~~I =
IHM,

IAOI==-R-> -R-=lOBI = IMOI •
Thus, I H K I > I H M I, and the point of inter-

section of the bisectors is found inside the triangle BOH.
342. Denote: IABI = IBel = a, IAMI = c,
IMCI=b, 1MB I==m, LBMO="P, LMBO=cp.
We have to prove that lOBI> IOMI or"l'>cp
or cos", < cos q>. By the law of cosines for the
triangles MBA and M BC, we get:
m 2-t·a2 - c2
2ma

cos q:> - cos '"

m 2+b2 - a2
2mb

m 2 (b-a)-a (b2 - a2)+b (a2 - c2 )
2mab

But a-e=b-a; hence,
(b-a) (m 2 - a b - a2+ab+bc)
cos cp - cos 'l' =
2mab
(b-a) (rn2 - 4 2 - b (24- b»
2mab

=

(b-a) (m+b-a) (m-a+b)
2mab

0

> ,

343. Through the point M, we draw a straight
line parallel to A C to intersect A B at. a point K.

366


We easily find: I AK I = I eM

IMBI·

::i',

I':~:II

.IMKI=

Since IAMI:;;;; IAKI + IKMI. re-

IAKI
and IKMI, we get IAMI ~
ICMI·IABI
IMBI·IAC( =}(IAMI-IACI) X
IBCI
ICBI
IBCI ~ (IABI- lAC!) IMCI, which was to be

placing

+

proved.
a2 b2 + C2
344. The minimum is equal to --3~-and is reached if M is the centre of mass of the
triangle ABC. (This can be proved, for instance,
using the method of coordinates or Leibniz ' theorem-see Problem 140 in Sec. 2).
345. Let us "rectify" the path of the ball. To
this end, instead of "reflecting" the ball from the

+

A

I

I
I
I
I
J

t

I
D
Fig. 61
side of the billiards, we shall specularly reflect
the billiards itself with respect to this side. As
a result, we obtain a system of rays with a common vertex; any two neighbouring rays form an
angle a.. The maximal number of rays in the system which can be intersected by a straight line
is just the maximal number of reflections of the

367


+

ball. This number is equal to [ : ] 1 if : is
not a whole number ([xl the integral part of the
number z}; if ~ is a whole number, then it is

a

equal to the maximal number of reflections.
346. If the roads are constructed as is shown in
Fig. 61 (A, B, C and D denote the villages, and
the roads are shown by continuous lines), then
their total length is 2
2
< 5.5. It is possible
to show that the indicated arrangement of the roads
realizes the minimum of their total length.
347. If one of the sides of the triangle through
A forms an angle q> with the straight line perpendicular to the given parallel straight lines, then
the other side forms an angle of 1800 - q> - ex;
on having found these sides, we get that the area
sin a
of th e tri
riangI ' equaI to
e IS
2 cos ab cos (cp
q>
ex
tal- sin ex
This expression is
cos a
~s (a
2q»
minimal if ctl
2q> = 1800 •

+ Y3

+ )

+

Answer:

+

+

$'ftiln =

ab cot

a.
'"2.

IABI

348. We
have:
SACBD= IMOI SOCD =
2 (k+1) SOCD. Consequently, SABCD is the greatest if the area of the triangle OeD is the greatest.
But OCD is an isosceles triangle with lateral side
equal to R, hence, its area is maximal when the
sine of the angle at the vertex 0 reaches its maximum. Let us denote this angle by q>. Obviously
q> ~ q> < n, where «Po corresponds' to the case
wten AB and CD are mutually perpendicular.
Consequently, if CPo ~ n/2, then the maximal area
of the triangle OCD corresponds to the value q>t =
n/2, and if <Po > n/2, then to the value fJ't = (J'o.
Answer: if s«; V2-1, then Smax=(k+1) R2;
if k ;» l/2-1, then Smax=2R2 Yk (k+2)/(k+1).

368


3i9. Let the straight line BC satisfy the condition: I BP I == I MC I (the order in which the
points follow is B, P, M, C) Weare going to
prove that the area of the quadrilateral ABNC
is the smallest. We draw another straight line
intersecting the sides of the angle at points B 1
and C t " Let the point B lie between the points A
and B h then the point Ct lies between A and C.
We have to prove that S BBIN > SCCIN0 This
inequality is equivalent to the inequality S BBtP >
.
SBBtP
SCCtP
IAPI
.
SCCIP' since S
8
= IANI Adding
0

RBIN

CCIN

8 sec, to both sides of the last inequality, we get:
S BBIP + 8 BPCt == S BBIPCI == 8CICBl
(follows
from the equality I BP I = I MC I) for the leftS BPC. == SC1CB for
hand member and SCCtP
the right-hand member. But, obviously, SCtCBI >
SCICB. The case when the point B 1 lies between A
and B is considered in a similar way.
Construction. It suffices to draw a straight line
to intersect the sides of the given angle and the
straight lines A N and A M at points B 0' Po, M 0'
and Co, respectively, so that I BoP o I = I MoC o I
and then to draw through M a straight line parallel to BoGoo Consider the parallelogram ABoDC o;
let K and L denote the points of intersection of
the straight lines A Po and AM 0 with BoD and CoD,
respectively. It follows from the equality
IBoP o I = IMoC o I that SABoK = SACOI... The
problem is reduced to constructing two equivalent
triangles ABoK and ACoL all of whose angles are
known. Taking B o arbitrarily, we construct the
triangle ABoK. We then take on ABo a point
E such that LBoKE = LALC o and construct
the line segment A Coequal to Y I B oE I IBoA I.
BoC~ is the required straight line.
Remark. Consider the following problem.
Through a point M lying inside a given angle draw

+

369


a straight line intersecting the sides of the angle
at points Band C so that the line segment BC
is the smallest. I t follows from the above problem
that BC will be the smallest line segment if
I BPI == I M C I, where P is the projection of
the vertex of the given angle on Be. (It follows
even a stronger assertion, namely, if the line segment BC possesses the indicated property, then
for any other straight line passing through M and
intersecting the sides of the angle at points B I
and C1 the projection of the line segment RIC I on
the line segment BC is greater than I BC l-) However, it is not always possible to construct such a
line segment by means of a pair of compasses and
a ruler.
350. Let M 1 and N 1 be two other points on the
sides of the angle (Fig. 62). Then LMIAN 1 = ~,

o

~.u....-.._""""'--~"'----_

Fig. 63

Fig. 62

LAM1M
360° - a - ~ - LON1A > 180° LON1A = LAN1N. Hence, bearing in mind that
LMAM 1 == LNAN 1 , we get that I MIA 1<
I N1A I, and, hence, SMIAM < SNt A N; thus,
SOMtANl

<

SOMAN·

351. Taking into account the results of the preceding problem, we have to find out on what conditions we can find on the sides of the angle points
M and N such that LMAN = ~ and I MA I ==
I A N I. Circumscribe a circle about the triangle
MON (Fig. 63). Since q>
"I' P< 180°, the

+ +

2~-01557

370

point A is located outside this circle. If L is the
point of intersection of the straight line OA and
the circle, then the following inequalities must be
fulfilled:

and

LAMN=90 0 -

LANM = 900

-

:

~ >
>

LLMN=LLON

LLOM.

Thus,

if

'P < 90 ~ and 1jl < 90 - ~ , then it is
possible to "find points M and N such that
I MA I == I AN I and LMAN = ~. If the conditions are not fulfilled, then such points cannot
be found. In this case, the quadrilateral of the
maximal area degenerates into a triangle (either
M or N coincides with 0).
352. Let us take a point Al on BC (Fig. 64).
The quadrilateral OM A IN is equivalent to the
0

0

lJ

o
Fig. 64
quadrilateral OMAN, LMA1N< LMAN; consequently, if we take on OB a point M 1 such that
LM1A1N = LMAN, then SOMIAIN> SOMAN;
hence, the area of the quadrilateral corresponding
to the point AI' which, taking into consideration
the results of the two previous problems, proves
the statement.
353. Let, for definiteness, sin a:>- sin~; on
the extension of AB, we take a point K such that
LBKC ==~. Since
LCBK = LADC (since

371

ABeD is an inscribed quadrilateral), the triangle
KBC is similar to the triangle ACD. But I BC I:>

I CD I, consequently,

S ABeD·

B t S

S ABCD ~

U

AKC

a 2 sin (a

=

SBCK ~SADC

+ p) sin a.
A.

2 .

and

SAKe ~

at sin (a+p)sina
h
2 sin ~
,ence,

SIn ...

..

.

In similar Iashion,
a2 sin (a, +~) sin ~

we can prove that S ABCD ~ ----:2~.---..;.---------
SID a.
354. Consider the other positions of the points
M t and Nt (LM1AN 1 = ~) and, bearing in mind
the condition ex
~ > 180°, show that the "added"
triangle has a greater area than the triangle by
which the area is reduced (similar to the solution
of Problem 350 of Sec. 2).
355. Taking into account the result of the preceding problem and reasoning exactly as in Prob-

+

lem 351 in Sec. 2, we get: if cp

> 90°- ~ and

'!' > 90° - ~ ,then a quadrilateral of the smallest
area exists and for it I M A I == IA N I. If this condition is not fulfilled, then the desired quadrilateral degenerates (one of the points M or N coincides
with the vertex 0).
356. We take the point A for which the conditions of the problem are fulfilled and some other
point A to Drawing through Al straight lines parallel to A M and A N and which intersect the sides
at points M 1 and N l' we make sure that
SOM1AINl < SOMAN and, consequently, the more
so, the area of the minimal quadrilateral corresponding to the foint A 1 is less than the area of
the quadrilatera OMAN which is the minimal
quadrilateral corresponding to A.
357. The radius of the largest circle is equal to
2RI Y3, that is, to the radius of the circle circumscribed about the regular triangle with side 2R.
(Let us take such a triangle and, on its sides as
2~·

372


diameters, construct the circles.) For any circle
of a greater radius, provided it is coverable by the
gi ven circles, there is an arc of at least 1200 covered
by one circle, but such an arc contains a chord
greater than 2R. Thus, we have arrived at a contradiction.
In the general case, if there is an acute triangle
with sides 2R It 2R 2' 2R 3' then the radius of the
circle circumscribed about this triangle is the required one. In all other cases, the radius of the
greatest circle is equal to the greatest of the numbers s.; R 2 , R 3 •
358. It is possible. Figure 65 shows three
unit squares covering a square 5/4 on a side.

Fig. 65
359. Let us first note that the side of the
smallest regular triangle covering the rhombus
wi th side a and acute angle of 60° is equal to 2a.
Indeed, if the vertices of the acute angles M and
N of the rhombus lie on the sides AB and Be of
the regular triangle ABC and LBNM = a, 30° ~
a ~ 90°, then, using the law of sines for finding
I BN I from the triangle BNM and I cs I from
the triangle KNC (K the vertex of the obtuse angle
of the rhombus which may be assumed to lie on
the side A C), we get after transformations: IBe I =

373


2a cos (6°;0-':- a) . Taking into account that 30o~
cos
a ~ 90°, we find that I BC 1:>- 2a. It is easy to
see that an equilateral triangle 3/2 on a side can
be covered by three regular triangles with side 1.
To this end, we place each of the unit triangles so
that one of its vertices is brought into coincidence
with one of the vertices of the triangle to be covered,
while the midpoint of the opposite side coincides
with the centre of the covered triangle.
Let us now show that it is impossible to cover
an equilateral triangle with side b > 3/2 with three
equilateral triangles of unit area. If such a covering were possible, then the vertices A, B, and C
would be covered by different triangles, and each
of the sides AB, BC, and CA would be covered by
two triangles. Let A belong to the triangle I, B
to the triangle I I, C to the triangle I I I, the centre
o of the triangle belonging, say, to the triangle I.
Let us take on A B and A C points Maud N, respectively, such

that IAMI

= IANI =

+

b. Since

= I cs I = ;

b > 1, the points M and
N also belong to the triangle I and, consequently,
the rhombus AMON is entirely covered by the
triangle whose side' is less than 21 A M I(21 A M I > 1),
which is impossible.

I BM I

.

360. Denote the ratios

:~;:

by a,

~,

IAMI
IMel '

ICNI
INBI and

and y. Then (see the solution of

Problem 221 in Sec. f) P == Qa~," S = Q (a+ f) X
1). Finally, take advantage of the
inequality (a
1) - (V ap1'+ 1)3.
36t. Let cot a=x, cotp=y, then cot''=
-xy+1
x2 + 1
z+y
. z+y - x , a2 cot a
b2cot~

<p + 1) (V +

+

+

+

+

374


e2 cot Y= (a 2 - b'l.- el ) x

x +1
+ c -+ .
x y
expression b (x + y) +

+b

2

2

2

(x+ y)

l
The minimum of the
2
1
c2 x +
+ with x fixed and x + y > 0 is reached
x y
for such an y for which the following equality is

fulfilled: b2(x+y)"=e2 x
•

c
Thus, -b =

Vx+y 1
x2 +

2

x

+ =
+1

y
sin y

·

sm

;'+Y _ b .
c

x'+1

n . Hence, the least
p

value of the given expression is reached for such
a, p, and y whose sines are proportional to the
sides a, b, and c, that is, when the triangles
under consideration are similar. But in this case
an equality occurs (it is readily checked).
362. Denote: p - a = x, p - b = y, p - c =
z (p the semi perimeter). Leaving 48 V3 in the
right-hand side of the inequality, we get, after
transforming the left-hand side (for instance,
a2 - (b - c)2 = 4 (p - b) (p - c) = 4yz) and replacing S by Hero's formula, the inequality xy
yz
zx:).. V 3 (x
y + s) xyz. Dividing both
sides of the inequality by V;Y% and making the
substitutions u = V (xy)/z, v = V (yz)/x, w =
V (zz)/y (x = UW, Y = vu, z = wv), we get the
inequality
u
v
w> V3 (uv vw wu),
which, on squaring, is reduced to the known in2
2
2 :>- uv
equality u
v
w
vw + wu.
363. There are two families of regular triangles
circumscribed about the given triangle (see Problem 305 in Sec. 2). On the sides of the triangle ABC,
we construct externally the triangles ABC lt BCA 1 ,
and CA Bland circumscribe circles about them.
The vertices of the triangles of the first family lie
on these circles (one per each circle). Let 1°2° 3
denote the centres of those circles (0 10208 is a
regular triangle, see Problem 304 in Sec. 2). The
triangle whose sides are parallel to the sides of

+

+

+

+ +
+ +

+

+

+

°

375


°

the triangle 1° 2 ° 3 has the greatest area (the secant passing through the point of intersection of
the two circles has the greatest length when it is
parallel to the line of centres; in this case its length
is twice the distance between the centres). The
area of the greatest triangle is 8 0 == 48010203 =

V3
-3-

(a + 2b + e + 28 wr-) ,where S is the area
y 3
2

2

2

of the given triangle (see the solution of Problem
305 in Sec. 2). The area of the greatest triangle
belonging to the second family is less. Among
the regular triangles inscribed in the given one,
the triangle whose sides are parallel to the sides
of the greatest circumscribed triangle has the
smallest area. This follows from the result of Problem 241 of Sec. 1. Its area is equal to 8 1 = 8 2/ So.
Thus, the area of the greatest circumscribed regular
11 3 (a2
b2
e2 )
triangle is So =
6
28, and
the area of the smallest inscribed triangle equals
8 1 = ~2 ,where 8 is the area of the given triangle.

+ +

+

361 Circumscribe a circle about the

triangle

AMC. All the triangles AtMC obtained as M
displaces in the arc A C are similar, consequently,

the ratio

II~~II

is the same for them. Therefore,

if M is the point of minimum of the expression
IBM 1·1 CM I
/ (M) =
IAtMI
,then BM must pass
through the centre of the circle circumscribed about
the triangle AMC, otherwise we can reduce IBM'
leaving the ratio

I';::iI u~changed.

Let now B 1

and Ct be, respectively, the points of intersection
of the straight lines BM and CM with the circle
circumscribed about the triangle ABC, then

376

IBMI·ICMI
IA1MI

I CM 1·1 AM I

IB1MI

IAMI·IBMI
ICIMI
lines A M and C M

Consequently, the straight
must also pass through the centres of the circles
circumscribed about the triangles B M C and
AMB, respectively. Thus, the point M is the
centre of the inscribed circle (see Problem 125 of
Sec. 2). In addition, in this case A 1 is the centre
of the circle circumscribed about the triangle
CMB, sin LMBC =

I;BI

sin

~BC

IBMI·ICMI
IAIMI
=== 2r.

-

21 A 1MI; hence,
Let us return to the question of the least value
for the function f (M). One of the theorems of
mathematical analysis states that a function,
continuous on a closed set, always reaches its
greatest and least values on that set. In particular,
this theorem is true for a function of two variables defined on a polygon. But the theorem is not
applicable directly to this problem, since the
function f (M) is not defined at the vertices of
the triangle ABC. But cutting away from the
triangle its small corners, we get a hexagon on
which / (M) becomes a continuous function and
has, consequently, its least value. It is possible
to prove that near the boundary of the triangle
f (M) > 2r. Therefore, if the cut-away corners
are sufficiently small, then the function / (M)
reaches its least value on the hexagons, and hence,
on the triangle, when M is the centre of the inscribed
circle, this least value being equal to 2r. On the
other hand, the function f (M) does not attain
its greatest value although it is bounded. Prove
that / (M) < I, where l is the length of the greatest
side of the triangle ABC, for all the points of the
triangle with the exception of the vertices, and that
f (M) can take on values arbitrarily close to I.
365. On the rays MB and MC, we take points
C1 and B I , respectively, such that I MC I I =


377

I MC I, I MB I I = I MB J (the triangle MC1B l
is symmetric to the triangle MBC with respect
to the bisector of the angle B M C), C~ and B 2 are
the projections of C l and B 1 on the straight line
AM, respectively. We have: I BM I sin LAMC
ICMI sin LAMB= IBIMI sin LAMC+ lelMI X
sin LAMB = I BIB" I + I CtC, 1:>- I BtC I I = a.
Writing two more such inequalities and adding
them together, we prove the statement of the
problem. It is easy to check that if M coincides
wi th the centre of the inscribed circle, then the
inequality turns into an equality.
366. (a) Let us first solve the following problem. Let M he a point on the side A B of the triangle ABC; the distances from M to the sides Be and
A C are equal to u and v, respectively; hI and h s
are the altitudes drawn to BC and A C, respectively.

+

Prove that the expression ~+~ reaches the
u
v
least value when M is the midpoint of AB. We
denote, as usually: I BC I == a, I A C I = b, S
the area of the triangle ABC. We have: au.+ bv =
·
.
28 , v = 28 -b au • Su bsti
stituting v Into t he expression

~+~==t, we get: atu 2-2Stu
u
v

+ 2hI S == O.

The discriminant of this equation is nonnegative,
S2 (t 2 4t) :>- 0, whence t:>- 4. The least value
t = 4 is reached for u == Sia = hJ/2, u = h 2/2.
It follows from this problem that the least value
of the left-hand member of the inequality of Item
(a) is attained when M is the median point. The
inequalities of Items (b) and (c) are proved in a
similar way. In Item (b) we have to determine for
what point M on the side AB the product uu reaches
its greatest value. In Item (c), we first divide both
sides of the inequality by uvw and solve the problem on the minimum of the function (hl/u - 1) X
(h,/v - 1) for the point M on AB.
367. Let for the acute triangle ABC the inequal-

378

ity I AC I ~ I AB I ~ I BC I be fulfilled; BD the
altitude, 0 the centre of the circumscribed and I
the centre of the inscribed circle of the triangle
ABC, E the projection of Ion BD. Since I ED I =
r, we have to prove that I BE I:> R = I BO l.
But BI is the bisector of the angle EBO (BI is the
bisector of the angle ABC and LABD = LOBe),
LBEI = 90°, LBOI"> 90° (the latter follows
from the fact that the projection of CIon BC does
not exceed IBCI/2). Consequently, IBEI:>- I BO I
(we map BO symmetrically with respect to BI).
368. Since the area of the triangle formed by
the medians of the other triangle is 3/4 of the
area of the original triangle, and for any triangle
abc = 4R S, we have to prove that for an acute
triangle the following inequality holds true:
5
mambmc>

8

(t)

abc.

Let, for the convenience of computations, one of
the sides be equal to 2d, and the median drawn
to this side be m, Since the triangle is acute-angled,
we have: m > d. Let t denote the cosine of the
acute angle formed by this median and the side
2d, 0 ~ t < dim (t < dim is the condition for
a triangle to be acute-angled). Expressing the
sides and median in terms of d, m, and t and
substituting the found expressions into the
inequality (t), we get after transformations:
m2 (9d2
m 2 ) 2 - 25d2 (d2 + m2)1 > t 2d2 m 2 (64m2 tOOd2). The left-hand member of the inequality
is reduced to the form: (m 2 - 4dm
5d2) X
(m 2
4dm
5d2) (m2 - d2). For m > d this
expression is positive. In addition, if m = d
(the triangle is right-angled), then the left-hand
member of the inequality is no less than the
right-hand member (equality for t = 0). Further,

+

+

+

+

if d < m ~ : s, then the right-hand member
of the inequality is nonpositive, and the inequality

379


i

holds true. Let m >
d. In this case, the righthand member of the inequality is less than the
value obtained for t == dim. But for t == dim the
original triangle is right-angled, and for right
triangles the validity of a slack inequality has
been already proved. (It suffices to repeat the same
reasoning with respect to the other side of the
triangle.) Thus, it has been proved that the inequality (1) is valid for any nonobtuse triangles
except for isosceles right triangles; for the latter
an equality occurs.
369. Let M lie inside ABC at distances z, y,
and %from the sides BC, CA, and A B, respectively.
The problem is to find the minimum of Z2 + y2 +
.;2 provided that a% + by + cz = 2S A B C • Obviously, this minimum is reached for the same values
of x, y, .; as the minimum of z2 + y2 + z2 21 (a%+ by +cz) == (x- )"a)2+ (y-1b)2+ (Z-Ac)2).,2 (a2
b2
c2 ) , where 1 is an arbitrary fixed
number (also provided that az by Col= 28 A Be).

+ +

+ +

Taking A= a l ~s
c l (A is found from the
equations % = Aa, y = )"b, z = Ac, ax + by +
cz = 2S A B C ) , we see that the minimum of the
last expression is reached for % = ).,a, y = ).,b,
z = Ac. Let now the point M be at distances Aa,
).,b, and Ac from Be, CA, and AB, respectively,
and the point M 1 symmetric to M with respect
to the bisector of the angle A. Since S AMIC =
S AM IB' M I lies on the median emanating from
A, and this means that J! lies on the symedian
of this angle (see Problem 171 in Sec. 2).
370. Let M be a point inside the triangle ABC
whose greatest angle is less than 120°. We rotate
the triangle AMC about the point A through an
angle of 60° externally with respect to the triangle
ABC. As a result, the point C goes into the point
Ch and the point M into the point Mil The
sum I AM I + IBM I + ICM I is equal to the

:SB.+

380


broken line BMMtC. This line is the smallest
when the points M and M t lie on the line segment
BC l.: Hence, there follows the statement of the
problem.
371. Let ABC be the given acute triangle, Al
a point on the side BC, B I a point on the side CA ,
Ct a point on the side AB; A and A 3 points symmetric to At with respect to the sides AB and AC,
respectively. The broken line A sCtBtA 3 is equal
to the perimeter of the triangle AIB1Ct ; consequently, with the point At fixed, this perimeter
is the smallest and equa s I A 2 A 3 I when the
points C1 and B 1 lie on the line segment A sA 3.
But AA sA 3 is an isosceles triangle, LA 2 A A 3"=
2 LBAC, I AlA I = I AlA I = IAA t ,. Hence,
I A,A s I is the smallest i AA 1 is the altitude of
the triangle BAC. In similar fashion, BB t and ce l
must also be altitudes.
372. If the greatest angle of the triangle is
less than 1.20°, then the sum of the distances takes
on the least value for the point from which the
sides can be observed at an angle of 120° (see
Problem 370 in Sec. 2). This sum is equal to I BC. I
(using the notation of Problem 370 of Sec. 2). The
square of this sum is equal to a 2
b2 1
~;2ab cos (LC+600) = 2 (at
b"
c2 )+ 2S JI 3.

+

+ +

But it follows from Problem 362 of Sec. 2
that a l
bl
cl ~ 48 V3. It remains to prove
the inequality S ::.>- 3 V3r 2 • It is proved in a rather
simple way; it implies that among all the triangles
circumscribed about a given circle the equilateral
triangle has the smallest area (for this triangle the
equality is fulfilled). To complete the proof, it
is necessary to check whether the inequality
a
b:>- 6r is true, since for a triangle with an
angle exceeding 120° the least value is reached by
the sum of the distances to the vertices at the vertex
of the obtuse angle.

+ +

+

381

Answers, Hints, Sotutlons

373. Let us prove the right-hand member of
the inequality. Let, for definiteness, b:>- e.
(1) If a ~ b, then 2p=a+b+e=(b-a)+e+
be
be+a'
2a<2e+2a~2-+2a=2 - - .
a

(2) If

a~b~c,

a

then a<2b and 2p=a+b+

2be
be+al
c=(b+c-a)+2a ~ c+2a<--+2a=2 - a .
II

The left-hand member of the inequality follows
from the right-hand member and the identity
(b+c)(p-a)-bccosA=a (

374. We have:

'

IBNJ

INCI =

bc+al
a

P

IAMI
IMCI =

)

IALI

ILDI

==:

I~~II that is, KN is parallel to CD, the quadrilateral KLM N is a parallelogram. Let I AK I = a,
a

%

IKCI=b, IBK!=x, IKDI=Y, y;;:::l); then

SKLM=SALM-SAKL=

x

X

• ( z+y -

(z~y )2 SADC

a

z+y • a+b

a)

a+b

x
SADC=

x+y

y
%Iy
y+z SABCD< (x+y)3 SABCD·

We denote: ylx = t, It is easy to prove that the
greatest value 4/27 is attained by the function
t/ (1
t)S for t = t/2 (for instance, by taking
the derivative of this function). Thus, SKLMN =

+

2SK L M

< 2~

SA sco-

375. Let at band c denote the sides of the triangle A Be t I the centre of the inscribed circle. The
following vector equality holds true (it follows

382


from the property of the angle bisector, see Problem
9 in Sec. 1):
-+

-+-

~

IA·a

+ IB·b + IC-e =

O.

(1)

In addition, I IB I < c, I tc I < b. These inequalities follow from the fact that the angles AlB
and AIC are obtuse. Let us take a point Al sufficiently close to the point A so that the inequalities are fulfilled as before: I ItB I < c, I le I <
b where II is the centre of the circle inscribed in
the triangle A tBC. The sides of the triangle A IBC
are equal to at bl , CI. The same as for the triangle
ABC, we write the equality
t

-+-

~

~

I 1A t·a+I tB.b 1 +/ t C .cl =0.
Subtract (1) from (2):
--+-

a (I lA t

-.

--+-

(2)

-.

--+-

~

I A)+ IIB·b t - I B·b+ / lC·Ct-1C.c= O.

-

(3)

Note that
-..

-+-

--+-

--+-

I 1A1 - I A = / 1/+AA t ,
-..
-+
--+-+
I IB.bl- IB·b-/tB (b1-b)+I.I·b,
--....

-+

~

I Ie ·CI- IC ·c+ I Ie (CI- c)

+

(4)
(5)

--+

11/ -c,
(6)
Replacing in (3) the corresponding differences
by the formulas (4), (5), (6), we get

--+-+--+
I II (a+ b+c)+AA 1 -a+liB (b l

-

b)
--+

+/tC (Cl- c)=O.
-+

-+

Since 1/1 BI <c, IItCI

I Cl -

cl

--+

fAAt·a

~

< IAIAI,

+

-+
I 1B(b l -

<b,

we have:

Ibl-hl

I/t/l

+ I1C(Cl -

1

a+b+c

-+

b)

< IAIAI,

e)]

<

X

IAAtl X

383


a+b+c == IAA11,
a+b+c

whence we can derive the
statement of the problem for any position of AI.
Remark. We have actually differentiated the
equality (1) and proved that I V A I > I VII, where
V A and VI are the velocities of displacement of
the points A and I, respectively.
376. Circumscribe circles about the triangles
ABF, BCD, and CAE. They have a common point
M. Since the angles of the triangle DEF are constant, LD = 'V, LE = a, LF == 6, the constructed circles and point M are independent of
fP. The side DF (and, consequently, EF and ED)
is the smallest when DF is perpendicular to BM.
Let
be the angle corresponding to this position.
"hen LMBC == LMCA = LMAB = 900 - CPo.
Extend CM to intersect the circle circumscribed
ahout the triangle AMB at a point Fl. We can find
that LF1BA == e , LF1AB = ~; FIB turns out
to be parallel to A C. From F I and B, we drop
perpendiculars FIN and BL, respectively, on A C.
Since I FIN I = I BL I, we have: tan «Po =
o
ICNI
IANI
IALI
cot (90 - «Po) == IF1NI =~ IFINI
IBL'I+
ICLI
l'BLT=cot p + cot ex + cot y. Thus, tan CPo =

'0

+

+

+

cot a
cot P cot y. Remark. The angle (a) ==
90° - CPo is called the Brocard angle, and the
point M the Brocard point. There are two Brocard
points for each triangle. The position of the second point M I is determined by the condition:
LM1BA = LM1AC = LMICB.
IACII
'BAli
ICBll
377. Set: l'AliI=x, lBCI=y, tCA! =z.
We _assume that x ~ 1/2. Suppose that the areas
of the triangles ABle l , BCIA lt and CAIB 1 are
greater than the area of the triangle AIBtC l • Then
z ~ 1/2 (otherwise S ACIBI ~ S AtCIBI) and y ~ 1/2.
The areas of all the triangles under consideration
are readily expressed in terms of SABe and x , y,

384

z, for instance: S AB Ct = x (1 -

s) SABe- The
inequality S AIBICl <ISABICI is reduced to the
form 1 - x (1 - s) - y (1 - x) - z (1 - y) <
x (t - z) _ Adding three such inequalities together, we get:
3 - 4x (1 - s) - 4y (t - x) 4z (1 - y) < o. The last inequality is linear with
respect to x, y, s , If it were fulfilled for certain
%, y, z between 0 and 1/2, it should also be fulfilled
for a set of the extreme values of the variables,
that is, when each variable is equal ei ther to 0 or
1/2. But it is possible to check to see that this
is not so. The obtained contradiction proves our
statement.
378. Let Q denote the midpoint of OH. As is
known, Q is the centre of the nine-point circle
(see Problem 160 in Sec. 2). We have: I OH 12
4 1
12 = 2 I 01 12
2 I HI 12 • Since I Qf 1 =
R/2 - r (by Feuerbach's theorem, Problem 287
of Sec. 2), I 01 12 = R2 - 2Rr (Euler's formula,
Problem 193 of Sec. 2), and bearing in mind that
R:>- 2r, we get:
1 OH 12 = 2 I IH 12
R2 - 4r2 :>- 2 I IH 12 •
379. An elegant idea for proving inequalities
of such a type was suggested by Kazarinoff (Michigan Mathematical Journal, 1957, No.2, pp. 97-98).
Its main point consists in the following. Take
points HI and CIon the raysAB and A C, respectively. It is obvious that the sum of the areas of the
parallelograms constructed on AB I and AM and
on AC1 and AM is equal to the area of the parallelogram one of whose side is BIC l , the other being
parallel to A M and equal to 1 AMI (see also
Problem 40 of Sec. 2). Consequently,
I AC1 I v
1 AB 1 I w ~ I B 1C1 1 x,
(1)
(a) Let us take the points B 1 and Cl coinciding
with the points Band C; then the inequality (1)
cw ~ ax. Adding toyields the inequality bv
gether three such inequalities, we get the required
inequality.

+

+

ot

+

+

+

385


(b) If I AB I I = I' AC I, I AC 1 I = I AB I, then
the inequality (1) will yield co + bui ~ ax or x:>
~ v+~ w. Adding together three such inequalia
a
ties, we get:

x+y+z;;" ( :

+ ~ ) u+ ( : + ; ) v
'b
a
+t-a+T )

w~2(u+v+w).

(c) In Item (a), we proved the inequality ax

bv+cw, whence xu
.

a

~ ~ uv+"':- wu. In
a
a

~

similar

cab

c

Iashion, yv ~ b uV+ b WV, zw ~ C uw -tVID.
Adding together these three inequalities, we get:

+ :) uv+(++ ~ ) vw+
( ; + : ) wu >2 (uv+vw+wu),

xu+yv+zw> ( :

(d) Let Alt BIt and C1 denote, respectively, the
projections of the point M on the sides Be, CA,
and AB of the triangle ABC. On the rays MA,
MAl' MB, MB lt MC, uc., take, respectively,
points A', A~, B ', B;, C', C; such that I MA i X
I MAil = I MAli I MA; I = 1MB 1·1 MB' 1=

I MB I 11MB; I = I Me I · I MC' I = I Mel I x

1M C; I = d 2 • • It is possible to prove that the
points A I , B ', C' lie on the straight lines BiC;,
C;A~, A;B;, respectively, MA', MB', Me' being
respectively perpendicular to these lines. Thus,
in the triangle AiBiC;, the distances from M
• This transformation is called inversion. See
the Remark to the solution of Problem 240, Sec. 2,
and also Appendix.
1/ 2

25-01557

386


d2

to the vertices are equal to
and to the opposite sides to

U'V'
d2

d2

7'

-

y

,

W

dB
z

Applying the inequality of Item (b), we get the
required inequality.
(e) Let us take in the inequality (1) bl = CI = l;

then al=2lsin ;

. We have x;;;;"

.1 A (u+v).
281D

2

On having obtained similar inequalities for y
and a, and multiplying them, we get:
/1
zyz~
. A . B " C (u+v) (v+w) (w+u)
8S1D - S i l l - s l n -

222

R

=2r"(~+v)

(v+w) (w+u)

"t
" A
· B
" C
r
( th e equal 1 y sin T· sin T· sin T=4R was

proved when solving Problem 240 in Sec. 1 , •
)
(f) From the inequality of the preceding item it
R
follows: zyz ~ 2 2 yuv. 2 yrViD. 2 V iDU = 4R X
r

.

r

uvw.
(g) Dividing the inequality of Item (d) by the
inequality of Item (f), we get the required inequality.
Remark. In the inequality of Item (a), equality
is achieved for any acute triangle when M coincides
with the intersection point of the altitudes of the
triangle" In Items (b), (c), (d), and (g), equality
is achieved for an equilateral triangle, when M
is the centre of this triangle. In Items (e) and (f),
equality is achieved in any triangle, when M is
the centre of the inscribed circle.
380. Consider the class of similar triangles.


387

As a representative of this class we choose such
a triangle ABC in which t AB I == v, t BC I = u,
I ..t C I = 1, u ~ v ~ 1. Thus, to each class of
similar triangles there corresponds a point B inside the curvilinear triangle CDE, where D is the
midpoint of the arc A C, the arc EC is an arc of
the circle with centre at A and the radius of t ,
ED being perpendicular toAC (Fig. 66). The trianE

A

L

o

Fig. 66

gle A BD will be called a "left-hand" triangle, the
triangle BDC a "right-hand" triangle. Consider
the process described in the hypothesis; in doing
so, at each step we shall leave only the triangles
similar to which we have not met before. For each
triangle we shall take the representative of the
class described above. Let X, Y, Z be midpoints of
AR, DB, CB, respectively; m = I DB I, h the
altitude of the triangle ABC. For "right-hand"
triangles, the following three cases are possible.
25*

388

(1) u~ 1/2, m~ 1/2 or u~ m, 1/2~ m, that
is, the greatest is the side DC or BD. This case
occurs if B is located inside the figure DMFC,
where DM is an arc of the circle of radius 1/2
centred at the point C, FC the right-hand part of
the arc EC, I DM I = I MC I = 1/2, DC and FM
line segments, FM .1 DC. In this case, the arc
MC (centred at D) separates the domain for which
DC is the greatest side in the triangle DBC
from the domain for which the greatest side is
DM. In this case, the representative of the
triangle DEC has an altitude equal to 2h if D.C

is the greatest side, or
5
h
- - 2 V1
2

-

2~a:> 2 I D~412

= ql (h) h, ql (h)

>

1 if h <

h2

¥7/4.

(2) u > m, U > 1/2, v > 2m. Note that the
equality v = 2m occurs for the circle with diameter LC, where I A L I = 1/3. Inside this circle
v > 2m. This case takes place if the point B is inside the curvilinear triangle DKN (KN and ND
arcs, DK a line segment). Since the triangle DZC
is similar to the original triangle ABC, we consider only the triangle D ZB . I ts greatest side is
DZ equal to v12. Its representative has the altitude

h

equal to 4 (V/2)2

h"

h

h

7> I AB 2 12 ~ IABal" =

h
=q(h)h
5/9 ~ (4/3) ~119 __ h2
2
l'

q(h»1.
2

(3) U:>- 1/2, u=> m, v ~ 2m. In this case, the
greatest side in the triangle BZD is BD equal to
m; and there is no need to consider the parts of the
triangle BDC since the triangle BYZ is similar
to the triangle BDC, and the triangle DYZ is
similar to the triangle A BD (we do not consider
the triangle D ZC any longer).

389

For "left-hand" triangles, two cases are possible,
they are analogous to Cases 2 and 3 for "righthand" triangles.
(2') If B is inside the figure DKNC, then the
triangle DXB, congruent to the triangle DZB, is
left for further consideration; its representative
has an altitude no less than 92 (h) h.
(3') If B is outside the figure DKNC, then
further consideration of parts of the triangle A BD
is ceased.
Note that, with an increase in h, the coefficient
q2 (h) increases, while ql (h) decreases and becomes
equal to 1 at the point F, h = v'7/4. Let us take
points P and Q on FM and the arc FC, respectively,
sufficiently close to F. Inside the figure
B 1KNM PQB 4, the inequalities ql (h):>- qo, qs (h) ~
qo, and s« > t are fulfilled. Consequently, in all
cases the rate of increase of h is no less than qo,
and in a finite number of steps or for all the triangles under consideration either Case 3 will occur or
the vertex of the triangle will be located inside
the curvilinear triangle PFQ. The case when the
point B is inside the triangle P FQ involves no difficulties and is considered separately. In that case,
"right-hand" triangles should be considered. It
suffices to meet the condition I FP I ~ I FM I =

117-; 1/3 . In the triangle

BDC, the side BD

equal to m is the greatest, h 2 <: 7/t6. We can
show that to the representative of the class of
triangles similar to the triangle BD C, there will
correspond a point lying outside the curvilinear
triangle PFQ. And since the altitude is not decreased in this case, Case 3 will occur for both
parts of the triangle BD C. The proof of the first
part has been thereby completed.
The second part follows from the result of Problem 327 of Sec. 2 and also from the fact that all
the triangles which are considered after the first
division have a representative whose altitude is

390


no less than h, and, consequently, the smallest
1
angle is 'no less than LB..AC> 2 LB1AC,>

1
yLBAC.
381. Let us formulate and prove the result
obtained by M. D. Kovalev which is stronger
than it is required by the hypothesis. Among all
the convex figures covering any triangle with sides
not exceeding unity the smallest area is possessed
by the triangle ABC in which LA =60°, I AB 1=1,
and the altitude drawn to AB is equal to cos 10°.
t

The area of this triangle equals

~ cos tOO ~

0.4924.
(1) Note that it suffices to find a triangle covering any isosceles triangle whose lateral sides
are equal to I, the angle q> between them not exceeding 60°. This follows Irom the fact that any
triangle with sides not exceeding 1 can be covered
by an isosceles triangle of the indicated type.
(2) Let us prove that any isosceles triangle
mentioned in Item (1) can be covered by the triangle
A Be. We construct a circle of radius 1 and centred
at the point C. Let K, L, M, and N be the successive points of its intersection with CB, BA,
and AC (L and M are found on BA), LLCM =
LMCN = 20°. Hence, isosceles triangles with the
angle 0 ~ cp ~ 20° are coverable by the sector
CMN, whereas triangles in which 20° < cp ~ L C
are covered by the triangle A Be if the end points
of the base are taken on the arcs K Land M Nand
the third vertex at the point C. Let us now construct a circle of unit radius with centre at the
point A. This circle passes through the point B,
again intersects Be at a point P, intersects the
side AC at a point Q. We get: LPAB =
180° -2 LB < Le, since B is the greatest angle
of the triangle A Be. Hence, taking the vertex
of the isosceles triangle at the point A and the

391

end points of the base at the point B and the
arc PQ, we can cover any isosceles ·triangle for
which LC < «p ~ 60° (even 180° - 2 LB ~
«p ~ 60°).
(3) Let us prOVA that whatever the arrangement (in the plane) of the isosceles triangle DEF
in which LDEF = 20°, I DE I = I EF f = 1 and
the equilateral triangle XYZ with side 1, the
area of the smallest convex figure containing the
triangles DEF and'XYZ is no less than 0.5 cos 10°.
First note that the side of the regular triangle containing DEF is equal to

~3 cos tOo.

(The follow-

ing statement is true: if one triangle can be placed
inside the other, then it can be arranged so that two
of its vertices are found on the .sides of the larger
triangle. Weare not going to prove this general
statement. It suffices to check to see its validity
in the case when one of them is the triangle DEF,
the other being a regular triangle. This can be
done easily.) Now, consider the smallest regular
triangle X1Y1Z1 with sides parallel to those of the
triangle XYZ, and containing the triangles DEF
and XYZ. The side of ~ X1Y1Z1 is no less
than (2/ V3)cos 10°, and the altitude is no less
than cos 10°. The vertices of the triangle DEF
must lie on the sides of the triangle X1Y1Z1 not
containing the sides of the triangle XYZ. Consequently, the sum of the distances from the vertices
of the triangle DEF which are outside the triangle
X YZ to the corresponding sides of the triangle
XYZ must be at least cos 10°- J!3/2, and the area
of the smallest convex polygon containing the
triangles DEF and XYZ is no less than
O.5(cos tOO- V3/2) + J!3/4=O.5 cos 10°.
(M. D. Kovalev also proved that the smallest (by
area) convex cover found for triangles with sides
exceeding unity is unique.)

Appendix: Inversion
Definitions
Consider in the plane a circle a, of radius R
centred at a point O. For any point A, distinct from
0, let us define the point A' in the following way.
The point A' is located on the ray OA so that
lOA 1·1 OA I = R2. Thus, for all points in the
plane, except for the point 0, a transformation
is assigned which is called the inverston with respect
to the circle a. This transformation is also called
a symmetry with respect to a circle, the points A
and A' being said to be symmetric with respect to
the circle cx. (If a straight line is assumed to be a
circle of infinite radius, then the symmetry with
respect to a straight line can be represented as a
limiting case of symmetry with respect to- a circle.)
The point 0 is called the centre of inverston, the
quantity k = R2, the power of inversion. Obviously,
the points A and A' are interchanged: A goes into
A " and A ' goes into A. All the points of the circle
a, and only those points, remain fixed. The interior points of the circle cx become exterior, and vice
versa.
We can "supplement" the plane with a point
at infinity (00) and assume that as a result of the
inversion the point 0 goes into 00, and 00 into O.
Henceforward, the points into which the
points A, B, C, .•. go 8S a result of the inversion
are denoted by A', B', C',

Basic Properties of Inversion
Let us consider the basic properties of an inversion leaving the simplest and obvious properties unproved and outlining a scheme for reasoning
in the rest of the cases. (Completing the reasoning
with missing links, considering various configurations, as well as carrying out computations and
making drawings are left to the reader.)

Appendix: Inversion

393

t , A straight line passing through the centre
of inversion goes into Itself.
2. If- the points 0, A, and B are not collinear,
then the triangles OAB and OB'A' are similar. The
vertices A and B', B and A' are similar. In addition, I A'B' I = (k I AB 1)/ lOA I- lOB I·
Note that the last equality is also true if the
points 0, A, and B are collinear.
3. A 'straight line not passing through the centre of inversion 0, goes into a circle passing through
O. In this case, if l is a given line, A the foot of
the perpendicular from 0 on It then l goes into a
circle of diameter OA'.
Let us take an arbitrary point B on I. From
the similarity of the triangles OAB and OB'A'
(Property 2) it follows that LOB 'A ' = LOAB =
90°.
4. A circle CI) passing through the centre of
inversion 0, goes into a straight line perpendicular to the straight line passing through 0 and the
centre of the circle 6>.
5. If a straight line l and a circle (J) go' into
each other in an inversion with centre at 0, then
the tangent to (il at the point 0 is parallel to I.
6. A circle CI) not passing through 0 goes Into
the circle CD' which does Dot contaiD 0 either.
In this case, 0 is the external centre of similitude
of the circles (a) and 6>'.
To prove this property, let us draw a straight
line through 0 and denote by A and B the points
of its intersection with the circle (in particular,
we may assume A and B to be diametrically opposite points on e). Suppose that B lies on the line
segment OA. Then A belongs to the line segment
OB'. If C is an arbitrary point of the circle, then,
taking into account the similarity of ap~ropriate
triangles (Property 2), we have: LA C' B' =
LOC'B' - LOC'A' = LOBe - LOAC:::;:
LACB.
Since the number of intersection points of two
lines remains unchanged in inversion, we have:
26-015&7

394


7. Depending on the position of the centre of
inversion, two touching circles go into:
(8) two touching circles (if 0 lies on neither
of them);
(b) a circle and a line tangent to this circle
(0 lies on one of the circles, but does not coincide
with the point of tangency);
(c) a pair of parallel lines (0 coincides with
the point of tangency).

The Angle Between Circles
The angle between two intersecting circles is
defined as the angle between the tangents to the
circles passing through one of the points of, their
intersection. The angle between a circle imd a
straight line intersecting this circle is defined as
the angle between that line and the tangent to
the circle passing through one of the points of
intersection. Here, we may assume that the angle
between the lines does not exceed 90°.
Obviously, the choice of the point of intersection is of no importance, for determining the angle
between two circles. It is also obvious that the
angle between the circles is equal to the angle
between their radii drawn to the point of intersection.
8. The inversion retains the angle between
straight Hnes, I.e., the angle between straight
lines is equal to the angle between their images.
If the centre of inversion coincides with the
point of intersection of the lines, then the assertion is trivial. And if this centre does not coincide
with the point of intersection of ·.the lines, then
it follows from Property 5 and the definition of
the angle between two circles or between a circle
and a straight line.
9. In inversion. the angle between two circles
is equal to the angle between their images.
Consider the case when the centre of inversion
does not lie on given circles. Let A be one of the

/tppendix: Inversion

395

intersection points of the circles 001 and 002 , ~
and li the tangents to WI and 6)2' respectively,
passing through A. Let us also assume that the
centre of inversion 0 does not lie on the straight
lines
and
In the inversion with centre 0,
the circles (a)t and Cl)2 go into w~ and co~, respectively, and the lines '1 and 12 into the circles I; and
'i touching ro; and ro~ at the point A' of their
intersection (Property 7), that is, the angle between 1; and 12 is equal to the angle between ~~
and m;, and since the angle between 1; and ,; is
equal to the angle between 11 and It (Property 8),
the angle between (J) ~ and CO 2 is equal to the angle
between (0 and (Ot'
to. If the circles ex and CI) are orthogonal, that
is, the angle between them is equal to 90°, then
In io,:ersion with respect to ex the circle CI) goes
into itself. And conversely, if in inversion with
respect to the circle ex the circle CI) not coinciding
with ex goes into itself, then a and (a) are orthogonal.
Obviously, the last property is symmetric with
respect to a and CI). The radii of the circles a and
6) are, respectively, equal to the tangents drawn
from the centre of one circle to the other circle.
On the basis of Property 10, the inversion can
be defined in the following way. All the points
of the circle a go into themselves. If A does not
belong to a and does not coincide with its centre,
then the image of the point ..-i is represented by
the point A' which is the second point of inter.section of any two circles orthogonal to a and passing through A, Now, the sense of the synonymic
name for inversion-symmetry with respect to
a circle-becomes clearer. From this definition
and the property of inversion to preserve the "angle
between two Intersecting circles, it follows that:
t t , For any circle CIl and two points A and B
going into each other in the inversion with respect
to (0 their images in the inversion with respect
to the circle a whose centre does not belong to
(a) are represented by the circle ro' and polnta A'

't

26*

'2'


396

B' which go into each other in the inversion
with respect to (0'. If the centre of a, lies on 0),
then CJ) goes into the straight line l, and the
points A and B into the points A' and B', symmetric with respect to I.

80d

The Radical Axis of Two Circles
Solve the following problem.
Given two non-concentric circles 6)1 and (lls.
Find the locus of points M for which the tangents drawn to the circles (1)1 - and (llt are equal.
Solution. Let 01. and 0. denote the centres
of the circles Ctll and (1)2' Tl and TI their radii, Al
and A I the points of tangency, respectively. We
have IM01 12 - I MO" 12 = (I MAl 12
rl) 2
(I MA" 1
T~) = rf - rl. Thus, all the points
belong to one and the same straight line perpendicular to 0 101 - This line is called the radical axis
of the circles 6)1 and (I)". To complete the solution of the problem, it remains to determine which
points of the found line satisfy its conditions.
I t is possible to show that if the circles do not
.intersect, then all the points of the radical axis
.are suitable. If (1)1 and (1)2 intersect, then the radical axis contains their common chord; but all
the points of the common chord are not contained
in the required locus of points. Therefore, if (1)1 and
(1),1 touch each other, then the point of tangency
is' -excluded.
Consider the circle a with centre M on the
radical axis of the circles 0)1 and 0), and radius
equal to the length of the tangent drawn from
M to (01 or (1)1. (M is assumed to be located outside 0)1 and (1),-) The circle a is orthogonal to the
circles (01 and (01. Thus, the points of the radical
axis situated outside the circles which intersect
or touch each other constitute the locus of centres
of the circles orthogonal simultaneously to 0)1
and (0" and there is an inversion that carries
each ot them into itself.

+

+

Appendix: Inversion

397

Now, let us prove one more property of the
inversion.
12. If the circles WI and 002 do Dot Jntersect,
then there is an inventon carrying them into
eoncentric circles.
"
Let us take a circle a orthogonal to"~l and
ro2 with centre on the straight line I containing
the centres of rot and IDs. Since the circle~l and
(Ill do not intersect, such a circle a is ~tent.
Let 0 be one of the intersection pointsref the
circle a and the line I. In the-inversion with centre
0, the line t goes into itself, and the circle a into
the straight line p _ The lines I and p intersect and
are orthogonal to the circles co; and roi whieh are
the images of 0)1 and COt in the inversion with
respect to a. Hence it fol ows that the centres of
Q); and ro~ coincide with the point of intersection
of the lines I and p, that is, CIlI and ro' are concentric circles, (Prove that if a straight fine is orthogonal to a circle, then the former passes through
its centre.)
Here, we should like to note that any circle
orthogonal to the concentric circles 0)' and 00;
is a straight line, that is, a circle of infinite radius. Hence, in the inversion with respect to the
circle a all the circles, orthogonal to the circles
(1)1 and CJ)t must go into straight lines. Consequently, all the circles orthogonal to rol and IDs intersect
the line l at two fixed points.
13. For aoy two circles (&)1 and (1)2' there exists
at least one inversion which carries them into
each other. The circle defining this inversion
is called the middle circle of WI and CJ)I_
Theorem 13 should be formulated more exactly
in the Iollowing way. If 6)1 and w, intersect, then
there exist exactly two inversions in which 001
goes into W 2 t and vice versa. If W1 and CJ)B touch
each other or do", not intersect, then there is only
one such inversion.
Let us first consider the case of intersecting
circles CDt and CIl~. Apply an inversion I with centre

398


in one of the points of their intersection; a8 a result, (01 and IDs go into intersecting straight lines
11 and ll' The lines 11 and have two bisectors with
respect to which 11 and " are symmetric, Consequently (Property t 1), in the inversion I those
bisectors go into two circles with respect to which
(1)1 and 6) t are symmetric,
If ml and ID. do not intersect, then there is an
inversion I (Property i2) carrying them into concentric circles 00; and 6)g. Let 0 denote the centre
of m; and m;, and rl and r. their radii. Inversion
with respect to the circle a,' with centre at 0 and
radius './ TIT, carries 6>i and 6)2 into each other,
In the inversion I applied, the circle a' goes into
the required circle ex with respect to which 6>1
and (01 are symmetric.
To conclude this section, let us give the definition of the radical centre of three circles. Consider
three circles Cl)t, CI)~H and 6>3 whose centres do not
lie on a straight line. It is possible to prove that
three radical axes corresponding to three pairs
of those circles intersect at a point M. This point
is called the radical centre of the circles COt, CI) 2'
and Cila, The tangents drawn from M to the circles
<tll' (I)" and 6)3 are equal to one another, Hence,
there is an inversion with centre M that carries
each of the circles (01' (02' and (03 into itself.

't

Problems and Exercises
t , Find the image of a square in the inversion
with respect to the circle inscribed in the square.
2. Given a triangle ABC. Find all points 0
such that the inversion with centre 0 carries the
straight lines A B, Be, and CA into circles of
the same radius.
3. Let A', B', and C' denote the images of
the points A, B, and C, respectively, in the inversion with centre at a point o. Prove that:
(a) if 0 coincides with the centre of the circle
circumscribed about the triangle ABe. then the

.. ppendix: Inversion
A

399

·triangle A 'B'C' is similar to the triangle ABC;
(b) if 0 coincides with the centre of the inscribed
circle, then the triangle A'B'C' is similar to
the triangle whose vertices lie at the centres of
the escribed circles;
(c) if 0 coincides with the intersection point
of the alti tudes of the triangle ABe, then the
!l'iangle A'B'C' is similar to the triangle with
vertices at the feet of the altitudes of the triangle.
4. Points A and A' are symmetric with respect to a circle a, M is an arbitrary point of the
circle. Prove that t AM 1/1 A'M I is constant.
5. Two mutually perpendicular diameters are
drawn in a circle a. The straight lines joining the
end points of one of the diameters to an arbitrary
point of the circle a intersect the second diameter
and its extension at points A and A'. Prove that
A and A ' are symmetric with respect· to the circle a.
6. Prove that if a circle CJ) passes through the
centre of a circle a, then the image of CJ) in the
inversion with respect to a is their radical axis.
7. Given a circle and two points A and B on
it. Consider all possible pairs of circles touching
the given circle at the points A and B· and touching each other at a point M. Find the locus of
points M.
8. Given two touching circles. An arbitrary
circle touches one of them at point A and the other
at B. Prove that the straight line AB passes through
a fixed point in the plane. (In the case of equal
circles A B is parallel to the straight line passing
through their centres.)
9. Given three circles aI' a 2 , as, passing
through the same point. The straight line passing
through the points of intersection of the circles
I al and a 2 contains the centre of the circle as; the
straight line passing through the points of intersection (L2 and as contains the centre of the circle
at. Prove that the straight line passing through
the points of intersection as and at contains thQ
centre of the circle a 2 •

400


10. Given two circles (1)1 and ro2 • Consider
two arbitrary circles which touch the given circles
at some points and also each other at a point M.
Find the locus of points M.
11. Prove that by inversion any two circles
can be carried into two equal circles.
12. Prove that by inversion any four points
A, B, C, D, not lying on a straight line can be
carried into the vertices of a parallelogram.
13. The inversion with respect to a circle with
centre 0 and radius R carries the circle with centre
A and radius r into the circle of radius r' Prove
that r' = (rR2)/1l OA 12 - r 2 I.
t4. Four points A, B, C, and D are given in
a
plane.
Prove
that
I AB I· I CD I

I AD I· 1BC 1:>- lAC 1• I BD

~.

+

15. In a triangle ABC, the side AC is the
greatest. Prove that for any point M the following
inequality holds: I AM I
I CM I:> I BM I·
~ 16. Prove that all the circles passing through
a 'given fo.nt A and intersecting a circle <X at diametrical y opposite points contain one more
fixed point distinct from A.
t 7. Given four pointsx , B, C, and D. Prove
that the angle between the circles circumscribed
about the triangles ABC and BCD is equal to
the angle between the circles circumscribed about
the triangles CDA and DAB.
t8. A circle CI) passes through the centre of a
circle ct. A is an arbitrary point of the circle <I).
The straight line passing through A and the centre
of the circle a intersects a common chord of the
circles a and ro at a point A '. Prove that A and A'
are symmetric with respect to the circle <X.
19. Given two non-intersecting circles, which
do not contain each other, and a point A lying
outside the circles. Prove that there are exactly
four circles (straight lines can also occur among
them) passing through A and touching the given
circles.
20. Let s denote the area of the circle whose

+

401

Appendix: Inversion

centre is found at a distance a from the point o.
The inversion with respect to the circle with centre
o and radius R carries the given circle into the
circle of area a'. Prove that 8' = s- R4/(a2 - R2)2.
21. Given two circles tangent to each other.
Consider two other circles tangent to the given
circles and to each other. Let r and r I denote the
radii of the last two circles, and dJ and d l the distances from their centres to the straight line
passing through the centres of the given circles.
d,
dt
dl
Prove t hat - - - =2 or dt
2.

I

rt

rl

I

-,-=
rt
rl
I

22. Let 0)1 and CJ)t be two circles tangent to
each other. Consider the sequence of distinct circles
a o, at, a 2 , •• , an' ... , each of which touches
6)1 and 0)1' and, in addition, the circle ak +1 touches
the circle all' Denote the radii of the circles
a o, at, · .. , an, · · t by ro, rl t • . , rn , · · t
and the distances from their centres to the
straight line passing through the centres of 0)1
and (J)2 by el t , d1 ,
., d n ,
Express d n in
terms of rn if:
(a) do =·0 (this case is possible if 0)1 and CJ)2
touch each other internally);
(b) do

=

kro.

.

23. Let al and at denote two intersecting circles,
A and B the points of their intersection, 0) an
arbitrary circle touching (Xl and a , r the radius
of the circle (I), and d the distance from its centre
to the straight line AB. Prove that the ratio rld
can take on only two distinct values.
24. Given two non-intersecting circles at and
a 2 and a collection of circles CJ)1' O)~" •• , O)n'
touching al and ai' where (J)2 touches COt, 0)3 touches
6)2'
., (i)n touobes COn-t. We say that the system of circles (J)1' 6)". ., CJ)n forms a chain
if O)n and (i)t touch each other. Prove that if for the
circles at and at there exists at least one chain
consisting of n circles, then there are infinitely
many chains. In this case, for any point A on either
(11 or (XI there is a chain for which A is the point

402


of tangency of one of the circles of the chain.
25. Prove that if for the circles at and a. there
exists a chain of n non-intersecting clreles (see
the preceding problem), then (8 ± r)1 -- tJI =
4Rr tanl (nln), where Rand r are the radii of the
circles al and at and d is the distance between
their centres. (The minus sign is taken iF one circle
is located inside the other, and the pl us sign if
otherwise.)
26. Consider three circles each of which touches
three escribed circles of a triangle, each of those
circles touching QDe of the escribed circles internally
and the two other escribed circles externally.
Prove that the three circles intersect at one point.
270 Let dl , d l , . ., d n denote the distances
from a point M lying on the arc AlA n of the circle
circumscribed about the regular n-gon A IA I • • ,4 n
to the vertices A lt A t · , An· Prove that

_1_+

dl~' +

+_1-_=_1_

tJ.d a

d n -ld n
d1dn •
28. Let 4lt a., .
an-I' ao denote the sides
-, An-lAn, AnAl of the n-gon
AlAs, A,A s,
AlA"
., An; Pit P2'
Pn-l' Po the distances
from an arbitrary point M on the arc A n A t of tho
circle to the straight lines AlA 2' A .tA 3'
An-tAn; AnAl·
Prove that ~=..!!..-r-~ -t
+ ~•
Po Pl
PI
I' fa - 1
0

0'

0

-

•

Hints and Solutions
2. There are four points possessing the required
property: the centre of the circle inscribed in
the triangle and the centres of the three escrl bed
cireles.
3. (b) Prove that the triangles OAB and Olbla
are similar. Now from Property 2 it follows that
the straight lines A '8.1-/and I a1b are parallel.

Appendix: Inversion
7. Let a and ~ be circles touching the given
circle <I) at points A and B. In the inversion with
centre at A, the circles ID and a go into parallel
straight lines land p, the circle 8 into the circle 6'
which touches 1 at a fixed point h' and the straight
line p at a point M'. Thus, M' lies on the straight
line passing through B' perpendicular to I. The
required locus is a circle passing through A and B
and orthogonal to 00. (The points A and B themselves a e excluded.) Its centre is found at 'the point
of intersection of the tangents to CI) passing through
A and B.
8. Let 0 denote the point of tangency of the
given circles. In the inversion with centre at 0
those circles go into a pair of parallel straifht lines
containing the points A' and B', the ine segment A 'B' being perpendicular- to them. The
straight line A B goes into the circle circumscribed
about the triangle A'B'O; this circle, obviously,
passes through the point P symmetric to the point
o with respect to a straight line equidistant from
the obtained parallel lines.
9. Let 0 be the point of intersection of the circles (Zt, a 2,a.; and AI' AI, .A 8 , respectively, the
points of intersection, distinct from 0, of the circles a l and aa, a. and aI' (1,1 and ag. The inversion
with centre at D carries the circles ai' a 2 , Ct. into
the straight lines forming the triangle A ;A~A3.
From the hypothesis and Property 3 it follows
that AiD 1. A ~A~, A;O 1. AlA;. Hence, 0 is the intersection point of the altitudes of the triangle
A;A,Ai and A'O -L AiA;.
to. If (all and (Il, intersect, then the desired locus
consists of two circles-the middle circles (01 and
(01 (Theorem f3) excluding the
points of intersection of (Ill and (0, themselves. If they touch each
other, then it consists' of one middle circle, excluding the point of tangency. To prove this, It
suffices to apply an inversion with! centre at a
common point of the circles ID and 6)1' If Cl)l and
COs have no points in cOinmon" then the entire midI

dIe circle is the locus. In this case, we have to apply
the inversion carrying (1)1 and CJ)t into concentric
circles.
1t. Any inversion with centre on the middle
circle possesses the desired property since this
inversion carries the middle circle into a straight
line with respect to which the images of the given
circles are symmetric.
(1) The points A, B, C t and D lie on the same
circle (I). The given points may be regarded as the
successive vertices of the inscribed quadrilateral.
Let 0 be the 'point of intersection of the circle
orthogonal to (I) and passing through A and C with
the circle orthogonal to ID and passing through B
and D, In the inversion with centre 0 the quadrilateral ABeD goes into the inscribed quadrilateral A 'B'C'D' whose diagonals are diameters, .that
is, A 'B'C'D' is a rectangle.
(2) At B, .C, and D do not lie on the same-circle.
Let IDA' CI)B' COe, Cl)D denote the ~~Jcles-circumscribed
about the triangles BCD;- CDA, DAB, ABC,
respectively. We take the middle circle for (J) B
and (OD separating the point B from the point D
and tlie middle circle for IDA and IDc separating
.the points A and C. Let 0 denote the point of
their intersection. (Prove that those circles intersect.) In the inversion with centre 0, the given
points go into the vertices of a convex q.uadrilateral A 'B'C'D' each of whose diagonals separates
it into two triangles with equal circumscribed
circles (see Problem 11); consequently, the opposite
angles of the quadrilateral are equal, hence it follows that A 'B'C'D' is a parallelogram (prove it).
t3. Let the line OA intersect the circle with
centre at A at points Band C. Then' B' C' I = 2r'
Now, we can use the formula given in Item 2.
14. We afply the inversion with centre at A'.
We have I B C' I + I C'D' 1:>- I B'D' I. Then use
the formula given in Item 2.

Appendix: Inversion

405

15. It follows from the preceding problem that

I AC 1·1 BM , ~ I AB I- I cu I + I BC I· lAM I.

Since A C is the largest side, I BM I =0;;; II~ : '

IBCI
I eM 1+ I AC I · lAM I ~ I AM I + ·1 Me I.·
J

16. Let A' be obtained from A by inversion
'lith respect to the circle a; A 1 is symmetric to A '
about the centre of the circle ct. Prove that all the
mentioned circles pass through AI.
17.We apply the inversion with centre at A.
The first angle is equal to the angle between, the
straight line B'C' and the circle' circumscrib~d
about B'C'D', the second-to the angle between
the lines D'C' and D'B'.
18. The inversion with respect to the circle
a carries the straiflht line AB into 6).
19. We apply .ihe inversion with centre at A.
Then the statement of the problem is equivalent
to the statement that two circles arranged outside
each other have exactly four tangent lines.
.
20. Let the straight line passing through the
centre of the inversion and the centre of the given·
circle intersect the given circle at points whose
coordinates are %1 and %1 (the origin lying at the
•
point

n

T tX l -

0).

Then
R4

(RI

R')2 =

1t
s, = -4 - - - %1
%1'
R4

%1)' (%IX,)1 = s (41 - RI)I •
21. Note that in the inversion with centre at 0,
for any straight line t passing through 0 the f.llowing equality is true: dlr = d'lr' for an arbitrary circle, where rand r' are the radii of the
given circle and its image, respectively, d and
d' are the distances from their centres to the line
" respectively. This follows from the fact that
o is the external centre of similitude of both circles (Property 6).
"
Let us return to our problem, We apply the
inversion with centre at the point
tangency of

0'

406


the given circles. The given circles go .into a pair
of parallel straight lines, the line I passing through
the centres of the given circles is perpendicular
to them. The circles with the radii rl and r, go
into a pair of circles of the same radius r' which
touch each other and also a pair of parallel lines
obtained. Now it is obvious that if the centres of
the last two circles lie on the same side of I, and,
d~ di
f or defi m°t eness, d' > d'l' th en rrr::: di , 2r'
2
r

d'
+==2. If on both sides, then
r

r

+

r
d'
d'
--++4-=2.
r
r

22. Use the result of the preceding problem.
We get in Case (a) d n = 2nr n ; in .Case (b) two ank) r n and d n =
swers are possible: dn = (2n

I k - 2n I r n •

+

23. We apply the inversion with centre at A;
the circles a l and a l go into the straight lines
11 and
intersecting' at the point B' situated on
the straight line AB., As was proved when solving Problem 21, rid = r'ld', But r'ld'is the ratio of
the radius of the circle touching the lines 11 and
to the distance from its centre to the fixed straight
line passing through the point of intersection of
and I •. Hence, r'ld' takes on only two values
depending on which of the two pairs of the vertical
angles formed by IJ and I. the circle is located.
24. We apply the inversion carrying <Xl and as
into concentric circles (see Theorem 12). This done,
the assertion of the problem becomes obvious.
This theorem is called Steiner's porism,
25. If aJ and as are concentric circles with
radii Rand r, then the validity of the equality
(R - r)2 = 4Rr· tan2 (nln) (d = 0) is readily obtained from the obvious relationship R - r =
(R
r) sin (nln), R > r. We apply the inversion
whose centre is at a distance 4 from the common
centre of the circles al and at. Let, for definiteness,
a > R. The circles at and <XI will go into the
circles at and a;, ai inside a.i. In this case, by

'I

'2

'1

+

401

Appendix: Inversion

the formula from Problem 13, we have R' =
h p IS
:..a 2 RplRI , r' = a2 rpl rI ' wr ere i · the power 0 f
_
_
inversion. To find d' (the distance between the
centres of the circles al and at) we draw a straight
1ine through the centre of tile inversion and the
centres of (1,1 and (I, ; the segment of this line enclosed between the ~rst two points of Intersection
with the circles (1,1 and fl, is equal to the width of
the annulus (R - r). Tlie inversion carries ~
segment into the segment of length b =
(R-r)p2
::: (4-r) (a-R)

d' = I R' (~-r)

(see

r' -b 1 =

p2

I

consequently,

Item ·2),

I

Rpt

_

41 -

RI
a (R'- r 2 ) pi

rp'

al-r~

= (a2 - r t ) (at _ RI) • Further,
replacing R' and r' with the aid of the formulas
: (4 - r) (a - R)
.

,

,

(R-r)(al+Rr)p~

derived above, we get R - r = (al

-

r'> (a' _ RI) ·

We have to verify the validity of the equality
(11' - r')2 - (d')2 = 4R'r' tan' (n/n). Expressing
all the quantities entering this equality in terms
of R, r, a, and p and simplifying the result obtained, we lead to the equality (R - r)2 (a 2
Rr)2 -

(R -

r)2

(a2 -

a2 (R

+ r)2 =

4Rr (a 2

+

-

r 2) X

R 2) tan2 (n/n)'o But (R - r)I=4Rr tan (n/n).
Hence, we have to check to see that (a2
Rr)1 - a2 (R
r)2 = (a 2 - r 2 ) (a 2 - R2).
This
can be done easily.
The case a < R Is identic to the above. And
if r < a < R, then ex; and a; are located outside
@8ch other, and in the given formula the plus sign
should be taken.
26. We apply the inversion with centre in
the radical centre of the escribed circles in which
the escribed circles go into themselves. This inyprlion carries the straight lines containjpO' 6h~ sIdes
of the triangle into the cir~les--~oned In the

+

+

408
'hypothesis. All the three cilftles p,8SS .through the
radical centre of the e$Crib,e¢,eircles of thetriangle.
27. We apply the' inversion- with centre at M
and of power 1. A~ a.re.8u.l.t,the.points At, AI ...,
.A•. go. into the pOints A;, Ai, ... , 4.;: situated
on a straight Ilne, .~et the side of the n-gon be equal
to CJ. From the formula of Item 2 it follows that
"
1
t
I A;A·~ I = dla, a; I A~A3 I = drla a;

I A~_lA~ I = d t d 4; I A;A 1= d td 4. Subn-l n
1 n
stituting these expresSions into the obvious relationshlp I A;A n I = I AiAil
A~Ai I
IA~_IA':
we get the desired result.
28. We apply the inversion with centre at M.
The vertices of the given n-gon go into n points
lying on a straight line, and

n

+"

't

+ .. +

I A;A~ I = I AiA, I + IA,Ail +. ·· + I A~-lA~ I·
(.)
Let p' denote the length of the perpendicular from
the point M on the straight line A ;Ati. From the
similarity of the triangles AlMA land A ;MA t (Property 2) it follows that

:

~;~illl

~. IAiA~1 =

..!!. p'. Simil~lY,
PI

I AlAs 1 =

;: P'.

'
lA n-l A' I n -

tln-l
- - p' ,
Pn-l

I A;A nI=-!L p '
Po
Substituting these expressions into the relationship
(.) and redu~ing by p', we get the required equality.

Problems in-plane-geometry-Sharygin

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