Derivations of inscribed & circumscribed radii for three externally touching circles (Geometry of Circle)

Derivations of inscribed & circumscribed radii for three externally touching circles
Mr H.C. Rajpoot (B. Tech, Mechanical Engineering from M.M.M. University of Technology, Gorakhpur-273010 (UP) India)
Mr Harish Chandra Rajpoot Feb, 2015
M.M.M. University of Technology, Gorakhpur-273010 (UP), India
1. Introduction:
Consider three circles having centres A, B & C and radii respectively,
touching each other externally such that a small circle P is inscribed in the gap
& touches them externally & a large circle Q circumscribes them & is touched
by them internally. We are to calculate the radii of inscribed circle P (touching
three circles with centres A, B & C externally) & circumscribed circle Q
(touched by three circles with centres A, B & C internally) (See figure 1)
2. Derivation of the radius of inscribed circle: Let be the radius of
inscribed circle, with centre O, externally touching the given circles, having
centres A, B & C and radii , at the points M, N & P respectively. Now
join the centre O to the centres A, B & C by dotted straight lines to obtain
& also join the centres A, B & C by dotted straight lines
to obtain (As shown in the figure 2 below) Thus we have
In
⇒
√
⇒ √
√
Similarly, in
⇒
Figure 1: Three circles with centres A, B & C and
radii 𝒂 𝒃 𝒄 respectively are touching each other
externally. Inscribed circle P & circumscribed
circle Q are touching these circles externally &
internally
Figure 2: The centres A, B, C & O are joined to each other by
dotted straight lines to obtain 𝑨𝑩𝑪 𝑨𝑶𝑩 𝑩𝑶𝑪 𝑨𝑶𝑪

√ ⇒ √ √
√
Similarly, in
⇒
√ ⇒ √ √
√
Now, again in , we have
⇒
Now, taking cosines of both the sides we have
( ) ⇒
⇒ ( ) ( )
⇒
⇒ ( ) ( )
⇒
⇒ √( ) √( )
⇒ ( ) ( ) ( )
⇒

⇒
⇒ ( ) ( )
Now, substituting all the corresponding values from eq(I), (II) & (III) in above expression, we have
(√ ) (√ ) (√ )
(√ ) (√ ) ( (√ ) ,
(√ ) {(√ ) (√ ) }
( *
{ }
Now, on multiplying the above equation by , we get
{ }
⇒ { } { }
{ }
{ }
⇒ { }
{ }
{ }
⇒ {
}
{
}
{
}
⇒ { }
Now, solving the above quadratic equation for the values of as follows

√{ } { }{ }
{ }
√
{ }
√
⇒ (
√
( √ )
)
Case 1: Taking positive sign, we get
(
√
( √ )
) (
√
)
⇒
Case 2: Taking negative sign, we get
(
√
( √ )
) (
√
)
⇒
Hence, the radius of inscribed circle is given as
√
Above is the required expression to compute the radius of the inscribed circle which externally touches
three given circles with radii touching each other externally.
3. Derivation of the radius of circumscribed circle: Let be the radius of circumscribed circle, with
centre O, is internally touched by the given circles, having centres A, B
& C and radii , at the points M, N & P respectively. Now join
the centre O to the centres A, B & C by dotted straight lines to obtain
& also join the centres A, B & C by dotted
straight lines to obtain (As shown in the figure 3) Thus we have
In
Figure 3: The centres A, B, C & O are joined to
each other by dotted straight lines to
obtain 𝑨𝑩𝑪 𝑨𝑶𝑩 𝑩𝑶𝑪 𝑨𝑶𝑪

√ √ √
⇒ √
Similarly, in
√ ⇒ √
Similarly, in
⇒
√ ⇒ √
Now, again in , we have
⇒
Now, taking cosines of both the sides we have
( ) ⇒
⇒ ( ) ( )
⇒
⇒ ( ) ( )
⇒
⇒ √( ) √( )

⇒ ( ) ( ) ( )
⇒
⇒
⇒ ( ) ( )
Now, substituting all the corresponding values from eq(I), (II) & (III) in above expression, we have
(√ ) (√ ) (√ )
(√ ) (√ ) ( (√ ) ,
(√ ) {(√ ) (√ ) }
( *
{ }
Now, on multiplying the above equation by , we get
{ }
{ }
⇒ { } { }
{ }{ }
{ }
⇒ { } { }
{ }{ }
{ }

⇒
⇒ { }
Now, solving the above quadratic equation for the values of as follows
√{ } { }{ }
{ }
√
{ }
√
⇒ (
√
( √ )
)
Case 1: Taking positive sign, we get
(
√
( √ )
) (
√
)
⇒
Case 2: Taking negative sign, we get
(
√
( √ )
) (
√
)
(
√
)
⇒
Hence, the radius of circumscribed circle is given as
√
Above is the required expression to compute the radius of the circumscribed circle which is internally
touched by three given circles with radii touching each other externally.
NOTE: The circumscribed circle will exist for three given radii ( ) if & only if the following
inequality is satisfied

(√ √ )
For any other value of radius (of smallest circle) not satisfying the above inequality, the circumscribed circle
will not exist i.e. there will be no circle which circumscribes & internally touches three externally touching
circles if the above inequality fails to hold good.
Special case: If three circles of equal radius are touching each other externally then the radii of
inscribed & circumscribed circles respectively are obtained by setting in the above expressions
as follows
⇒
√ √ √
√
( √ )
( √ )( √ )
( √ )
(
√
*
⇒
√ √ √
√
( √ )
( √ )( √ )
( √ )
(
√
*
4. Derivation of the radius of inscribed circle: Let be the radius of inscribed circle, with centre C,
externally touching two given externally touching circles, having centres A & B and radii respectively, and
their common tangent MN. Now join the centres A, B & C to each other as well as to the points of tangency M,
N & P respectively by dotted straight lines. Draw the perpendicular AT from the centre A to the line BN. Also
draw a line passing through the centre C & parallel to the tangent MN which intersects the lines AM & BN at
the points Q & S respectively. (As shown in the figure 4) Thus we have
In right
⇒ √
√ √ √
√
In right
⇒ √
√ √ √ √
Figure 4: A small circle with centre C is externally touching two
given externally touching circles with centres A & B and their
common tangent MN

In right
⇒ √ √ √ √ √
From the above figure 4, it is obvious that now, substituting the corresponding values, we
get
√ √ √ ⇒ √ (√ √ ) √ ⇒ √
√
(√ √ )
⇒ (
√
(√ √ )
)
√
√ (√ √ )
Above is the required expression to compute the radius of the inscribed circle which externally touches
two given circles with radii & their common tangent.
Special case: If two circles of equal radius are touching each other externally then the radius of inscribed
circle externally touching them as well as their common tangent, is obtained by setting in the above
expressions as follows
√ √
⇒
5. Relationship of the radii of three externally touching circles enclosed in a rectangle: Consider any
three externally touching circles with the centres A, B & C and their radii respectively
enclosed in a rectangle PQRS. (See the figure 5)
Now, draw the perpendiculars AD, AF & AH from
the centre A of the biggest circle to the sides PQ,
RS & QR respectively. Also draw the
perpendiculars CE & CM from the centre C to
the straight lines PQ & DF respectively and the
perpendiculars BG & BN from the centre B to
the straight lines RS & DF respectively. Then join
the centres A, B & C to each other by the
(dotted) straight lines to obtain . Now, we
have
Now, applying cosine rule in right
⇒
Figure 5: Three externally touching circles with their centres A, B & C and radii
𝒂 𝒃 𝒄 𝒃 𝒄 𝒂 respectively are enclosed in a rectangle PQRS.

⇒ √ √ ( )
√
√
In right
⇒
√ √ √
⇒
√
In right
⇒
⇒
Now, by substituting the corresponding values from the eq(I), (II), (III) & (IV) in the above expression, we get
(
√ √
+
⇒ (
√
+
⇒ √
⇒ √
⇒ √ { } { }
⇒ √

⇒ √
Now, taking the square on both the sides, we get
( √ )
⇒ ( )
⇒
⇒
⇒ ⇒
⇒ √
Above relation is very important for computing any of the radii if other two are known for three
externally touching circles enclosed in a rectangle.
Dimensions of the enclosing rectangle: The length & width of the rectangle PQRS enclosing three
externally circles touching circles are calculated as follows (see the figure 5 above)
√ √ (√ √ )
(√ √ )
Thus, above expressions can be used to compute the dimensions of the rectangle enclosing three externally
touching circles having radii .
6. Length of common chord of two intersecting circles: Consider
two circles with centres and radii respectively, at a
distance between their centres, intersecting each other at the points
A & B (As shown in the figure 6). Join the centres to the point
A. The line bisects the common chord AB perpendicularly at the
point M. Let then the length of common chord . Now
In right triangle ,
√ √
Similarly, In right triangle ,
√ √
Now,
Substituting the corresponding values, we get
√ √
Figure 6: Two circles with the centres 𝑶 𝟏 𝑶 𝟐 and radii
𝒓 𝟏 𝒓 𝟐 respectively at a distance d between their
centres, intersecting each other at the points A & B
𝒂 𝒃 𝒄 𝒃 𝒄 𝒂 respectively are enclosed in a

Taking squares on both the sides,
(√ √ )
√
√
√
⇒
√
Hence, the length of the common chord of two intersecting circles with radii at a distance between
their centres is
√{ }{ }
| |
Special case: If then the maximum length of common chord of two intersecting circles
at a central distance √| |
Conclusion: All the articles above have been derived by using simple geometry & trigonometry. All above
articles (formula) are very practical & simple to apply in case studies & practical applications of 2-D Geometry.
Although above results are also valid in case of three spheres touching one another externally in 3-D geometry.
Note: Above articles had been derived & illustrated by Mr H.C. Rajpoot (B Tech, Mechanical Engineering)
M.M.M. University of Technology, Gorakhpur-273010 (UP) India Feb, 2015
Email:rajpootharishchandra@gmail.com
Author’s Home Page: https://notionpress.com/author/HarishChandraRajpoot

Derivations of inscribed & circumscribed radii for three externally touching circles (Geometry of Circle)

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