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/