Circles and tangents with geometry expressions

Circles and Tangents with Geometry Expressions

CIRCLES AND TANGENTS WITH GEOMETRY EXPRESSIONS ..................................................................... 1

INTRODUCTION ................................................................................................................................................. 2

Circle common tangents........................................................................................................................................................... 3
Example 1: Location of intersection of common tangents............................................................................................. 4
Example 2: Cyclic Trapezium defined by common tangents of 2 circles...................................................................... 5
Example 3: Triangle formed by the intersection of the interior common tangents of two circles................................. 7
Example 4: Locus of centers of common tangents to two circles .................................................................................. 8
Example 5: Length of the common tangent to two tangential circles ............................................................................ 9
Example 6: Tangents to the Radical Axis of a Pair of Circles ..................................................................................... 10

Arbelos ................................................................................................................................................................................... 11
Example 7: Various Circles in an Equilateral Triangle................................................................................................ 12
Example 8: Two circles inside a circle twice the radius, then a third .......................................................................... 14
Example 9: A theorem old in Pappus’ time.................................................................................................................. 19
Example 10: Another Family of Tangential Circles..................................................................................................... 22
Example 11: Yet Another Family of Circles ................................................................................................................ 25
Example 12: Archimedes Twins................................................................................................................................... 28
Example 13: Squeezing a circle between two circles................................................................................................... 29

1

C I R C L E S A N D T R I A N G L E S W I T H G E O M E T R Y E X P R E S S I O N S

Introduction
Geometry Expressions automatically generates algebraic expressions from geometric
figures. For example in the diagram below, the user has specified that the triangle is right
and has short sides length a and b. The system has calculated an expression for the length
of the altitude:

B

b

a·b
a ⇒ 2 2
a +b

C

D
A

We present a collection of worked examples using Geometry Expressions. In most
cases, a diagram is presented with little comment. It is hoped that these diagrams are
sufficiently self explanatory that the reader will be able to create them himself.

The goal of these examples is to demonstrate the sort of problems which the software is
capable of handling, and to suggest avenues of further exploration for the reader.

The examples are clustered by theme.

2

Circle common tangents
The following set of examples explores some properties of the common tangents of pairs
of circles.

3


Example 1: Location of intersection of common tangents
Circles AB and CD have radii r and s respectively. If the centers of the circles are a apart,
and E is the intersection of the interior common tangent with the line joining the two
centers, what are the lengths AE and CE?

B

D

r
s

A
F C

a·r a·s
⇒ ⇒
r+s r+s
a

E

How about the exterior common tangent?

B

r

D

s
A
C
F

a E
-a·s

-a·r ⇒ -r+s

⇒ -r+s | s<r

| s<r

4

Example 2: Cyclic Trapezium defined by common tangents
of 2 circles
Given circles radii r and s and distance a apart, what is the altitude of the trapezium
formed by joining the intersections of the 4 common tangents with one of the circles?

2·r·s
⇒
a
B

2·r·s
G ⇒
a

r D
I
s

A C

a H
E

J

F

Notice that this is symmetrical in r and s, and hence the trapezium in circle AB has the
same altitude.

5


Look at the ratio of the areas of the trapezia in the previous diagram:

2 2 2 2 2 2 2 2
2·r ·s· a -r -2·r·s-s +2·r ·s· a -r +2·r·s-s
z1 ⇒
2
a
2 2 2 2 2 2 2 2
2·r·s · a -r -2·r·s-s +2·r·s · a -r +2·r·s-s
B z2 ⇒
2
a

G

r D
I
s

A C

a H
E

J

F
z1 r
⇒
z2 s

Notice that the ratio of trapezium areas is the same as the ratio of radii.

6

Example 3: Triangle formed by the intersection of the
interior common tangents of two circles
Notice that if A is the area of the triangle formed by the centers of the circles, then area
STU is:

2rstA
(r + s )( s + t )(r + t )

D

s

r·s·t· a+b+c · a+b-c · a-b+c · -a+b+c P
⇒ C
2·(r+s)·(r+t)·(s+t)
I

G
O
c W

B a

K X

r
F
M H

t
A
J
Y E
Q

b

L

N

Notice that this ratio is independent of the size of a,b, and c.

7


Example 4: Locus of centers of common tangents to two
circles
W take the locus as the radius r of the left circle varies. The midpoints of all four
common tangents lie on the same fourth order curve

4 2 2 4 3 2 4 2 2 2 2 2 2 2 2 3 2
⇒ 4·X +8·X ·Y +4·Y -12·X ·a-12·X·Y ·a+a -a ·s +Y · 4·a -4·s +X · 13·a -4·s +X· -6·a +4·a·s =0

D
G

K
B
E s

r
L
A (0,0) (a,0) C

M

H
I

R
F
J

We can use Maple to solve for the intersections with the x axis:

> subs(Y=0,4*X^4+8*Y^2*X^2+4*Y^4-12*a*X^3-
12*a*Y^2*X+a^4-s^2*a^2+(4*a^2-4*s^2)*Y^2+(13*a^2-
4*s^2)*X^2+(-6*a^3+4*s^2*a)*X );
4 X 4 - 12 a X 3 + a 4 - s 2 a 2 + (13 a 2 - 4 s 2) X 2 + (-6 a 3 + 4 s 2 a) X

> solve(%,X);
1 1
a - s, a + s, a, a
2 2

8

Example 5: Length of the common tangent to two
tangential circles
A succinct formula:

⇒ 2· r· s
B

D
r

s

A C

9


Example 6: Tangents to the Radical Axis of a Pair of Circles
The radical axis of a pair of circles is the line joining the points of intersection. The
lengths of tangents from a given point on this axis to the two circles are the same.

4 2 2 2 2 4 4 2 2 2
a +4·a ·b -2·a ·r +r +s +s · -2·a -2·r
⇒
2·a F 4 2 2 2 2 4 4 2 2 2
a +4·a ·b -2·a ·r +r +s +s · -2·a -2·r
⇒
2·a
B

D
H
b

r
G

s
a

A C

E

10

Arbelos
A set of examples studying circles squeezed between other circles.

11


Example 7: Various Circles in an Equilateral Triangle
We look at the radii of various circles in an equilateral triangle:

A

G
3·a
18 F

E

a
a

a
2· 3 D

B
C

a

12

A
I
3·a H
54 G

F

E

a
a

D

B
C

a

What would the next length in the sequence be?

13


Example 8: Two circles inside a circle twice the radius,
then a third

D
H

2·r
3 r

G

C

E
B

r
2·r
A

F

14

And if we keep on going:

2·r r
3 3
F
H 2·r
J
11
L
I r
K 9
M N
G D

2·r
r
B

C
r

E

A

15


The general case looks like this:

1
1 1 -1 1
+ +2· +
2·r x 2 2·r·x
2·r

B

F
x G
A H

D

r
2·r

C

E

We can copy this expression into Maple to generate the above sequence:

> 1/(1/2*1/r+1/x+2*sqrt(-1/2*1/(r^2)+1/2/x/r));
1
1 1 1 1 2
+ + −2 +
2 r x r 2 xr

> subs(r=1,%);

16

1
1 1 2
+ + −2 +
2 x x
> f:=x->1/(1/2+1/x+sqrt(-2+2/x));
1
f := x →
1 1 2
+ + −2 +
2 x x
> f(1);
2
3
> f(2/3);
1
3
> f(1/3);
2
11
> f(2/11);
1
9
> f(1/9);
2
27
> f(2/27);
1
19
>
A little analysis of the series can lead us to postulate the formula 2/(n^2+2) for the n’th
circle:

Let’s feed the n-1th term into Maple:

> f(2/((n-1)^2+2));

17


1
3 1
+ ( n − 1 )2 + ( n − 1 )2
2 2
In order to get the expression to simplify, we make the assumption that n>1:

> assume(n>1);
> simplify(f(2/((n-1)^2+2)));
1
2
2 + n~ 2
We see that this is the next term in the series. By induction, we have shown that the n’th
2
circle has radius
2 + n2

18

Example 9: A theorem old in Pappus’ time
A theorem which was old in Pappus’ days (3rd century AD) relates the radii to height of
the circles in figures like the above:

2·r r
3 3
2·r
F
H 11
J

I L
K

4·r 12·r
G D 3 11

2·r

B r

4·r C
r
3

E

A

19


Applying the general model, we get a formula:

B

E
F
x

2·r

D
⇒ 2· x · 2·r-2·x

r
A C

Again, we can copy this into Maple for analysis:

2 x 2r−2x
> subs(x=2/(n^2+2),r=1,%);

20

1 4
2 2 2−
2 + n~ 2 2 + n~ 2
> simplify(%);
n~
4
2 + n~ 2
We see that the height above the centerline for these circles is the radius multiplied by 2n.

21


Example 10: Another Family of Tangential Circles

A C B

N
D M
F a
K
L 14
3·a
2 a J 3·a
E I
H 31
G
3·a
22
a
5
3·a
10

22

We can follow through a similar analysis to the previous section:

1
1 1 -2 1
+ +2· +
3·a x 2 3·a·x
3·a

F
G
x H
E
D

B

a

3·a
2
C

A

> f:=x->1/(1/3+1/x+2/3*sqrt(-6+3/x));
1
f := x →
1 1 2 3
+ + −6 +
3 x 3 x
> f(1);
1
4 2
+ I 3
3 3

23


> f(3/10);
1
5
> f(%);
3
22
>

> assume(n>1);
> simplify(f(3/((n-1)^2+6)));
1
3
6 + n~ 2
> simplify(%);

24

Example 11: Yet Another Family of Circles
We generalize the situation from a couple of examples ago. We look at the family
generated by two circles of radius a and b inside a circle of radius a+b:

a·b·(a+b)
a·b·(a+b) 2 2
a +a·b+b
2 2
4·a +a·b+b H

J
a·b·(a+b)
2 2
9·a +a·b+b
L I
G
a·b·(a+b) a+b
K F
2 2
16·a +a·b+b N a

M B
D

C

E
A

b

The pattern is pretty obvious this time: the radius of the nth circle is:

ab(a + b)
n a 2 + ab + b 2
2

25


To prove this, we derive the formula for the general circle radius x, and analyze in Maple:

1
1 1 1 1 1 1
+ - +2· - -
b x a+b b·x b·(a+b) x·(a+b)
H

J

x
I
G
a+b
F

B

E

A

b

Now we try feeding in one of the circle radii into this formula in maple (first making the
assumption that the radii are positive (along with n>1 for later use):

> assume(a>0,b>0,n>1);
> f:=x->1/(1/b+1/x-1/(a+b)+2*sqrt(1/(x*b)-1/((a+b)*b)-
1/((a+b)*x)));
1
f := x →
1 1 1 1 1 1
+ − +2 − −
b x a+b x b (a + b) b (a + b) x

26

> f((a+b)*b*a/(9*a^2+b*a+b^2));
 1 2 2
9 a~ + b~ a~ + b~ 1
1/
 b~ + ( a~ + b~ ) b~ a~ − a~ + b~


9 a~ 2 + b~ a~ + b~ 2 1 9 a~ 2 + b~ a~ + b~ 2 

+2 − − 
( a~ + b~ ) b~ 2 a~ ( a~ + b~ ) b~ ( a~ + b~ ) 2 b~ a~ 

> simplify(%);
( a~ + b~ ) b~ a~
16 a~ 2 + b~ a~ + b~ 2
Let’s try the general case, feeding in the formula for the n-1st radius:

> simplify(f((a+b)*b*a/((n-1)^2*a^2+b*a+b^2)));
b~ a~ ( a~ + b~ )
b~ a~ + a~ 2 n~ 2 + b~ 2
By induction, we have proved the general result.

27


Example 12: Archimedes Twins
The given circles are mutually tangential with radius a, b and a+b. Archimedes twins are
the circles tangential to the common tangent of the inner circles. We see from the
symmetry of the radius expression that they are congruent.

a·b
⇒
a+b
a+b
a·b
⇒
a+b
I

K

a

b
A
E
C

28

Example 13: Squeezing a circle between two circles
Take a circle radius 2a centered at (a,0) and a circle radius 4a centered at (-a,0). Now look
at the locus of the center of the circle tangent to both.

B

E

F t
4·a

2 2 2
72·a -8·x -9·y =0 2·a
A (-a,0) D
(a,0) C

It’s an ellipse. From the drawing we can see that the semi major axis in the x direction is
3a. What is the semi major axis in the y direction?

29

Circles and tangents with geometry expressions

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