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### Mutually Tangent Circles

```
Date: 05/22/2000 at 18:26:31
From: Brian
Subject: Mutually tangent circles and tangent lines

This math puzzle has stumped me. Please give me some answers or at
least some suggestions. The puzzle was given with a figure, so I know
my figure is accurate.

Given:

Circle A with radius a
Circle B with radius b
Circle C with radius c
Line S
All three circles are tangent to each other
No circle is on the interior of another circle
Line S is tangent to circles A, B, and C at points D, E, and F,
respectively
D-F-E

What is c (the radius of C) as a function of a and b?

The figure below is not drawn to scale (and the actual circles are
round).

XXXXX
XXX     XXX
XX          XXX
X               X
X                 X
X                   X
XX                    X
X                     X
X                       X
X                       X
oooo       X                        X
ooo    oo    X                         X
oo        o   X                         X
o           oo X                         X
oo            ooX            B            X
o              oX                         X
oo              oX                         X
o               oXX                       X
o       A       oXX                       X
o               o X                       X
o               o X                      X
o              o  X                    XX
o             oo...X                  X
o            o.   .X                XX
oo          o.. C ..XX             X
oo       oo  .   .   XX         XX
ooooooooo   ....     XXXXXXXXXXX
_____________D_________F___________E____________________

Thank you for your help.
```

```
Date: 05/23/2000 at 06:01:05
From: Doctor Floor
Subject: Re: Mutually tangent circles and tangent lines

Hi, Brian,

Thanks for writing.

Let P be the point of intersection of the line through C parallel to
DE with the segment AD.

Note that AP = a-c and AC = a+c.

By Pythagoras' theorem we find that CP = 2sqrt(ac), and thus
DF = 2sqrt(ac).

In the same way we find EF = 2sqrt(bc) and DE = 2sqrt(ab).

Since DE = DF + EF we can derive:

2sqrt(ab) = 2sqrt(ac) + 2sqrt(bc)

sqrt(ab) = sqrt(ac) + sqrt(bc)
sqrt(ab) = sqrt(c) [ sqrt(a) + sqrt(b)]

sqrt(ab)
sqrt(c) = --------------- [Take squares]
sqrt(a)+sqrt(b)

ab
c = -----------------
a + b + 2sqrt(ab)

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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