Tangent Line and Circles

Date: 04/05/99 at 12:33:37
From: Tom Hammond
Subject: Length of segment of line tangent to two tangent circles of 
different radii

Two circles of different radius are tangent to each other. A line is 
drawn that is tangent to both circles.

What is the length of the segment between the two points of tangency 
of the line and the circles?

My sophomore son and I have been trying to answer this. I have usually 
gotten through these pretty easily, but this one has us totally 
stumped. Thank you so much for your help!   

Tom

Date: 04/05/99 at 17:03:50
From: Doctor Peterson
Subject: Re: Length of segment of line tangent to two tangent circles 
of different radii

Hi, Tom. This is a nice little problem that involves some useful 
ideas.

Draw the two circles and their tangent, and the line joining their 
centers. Then draw a radius to each point of tangency, forming a 
trapezoid ABCD. The angles at C and D are right, so AD and BC are 
parallel. 

Draw BE parallel to the tangent, and think about the triangle ABE. You 
should be able to find a relation that will give you the length CD 
from the radii AD and BC.

             \
                   \    D
             ***********\
          ***          /***   \
        **            /    **      \    C
       *            E+       *     *****\
      *             /      \  *  **    /**   \
     *             /           **\    /   *
 ----*------------+------------*-----+-----*----
     *            A            **    B    *
      *                       *  **     **
       *                     *     *****
        **                 **
          ***           ***
             ***********

Let me know if you need more help.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 04/05/99 at 21:06:44
From: thammond
Subject: Re: Length of segment of line tangent to two tangent circles 
of different radii

Thanks for the help, but I've run up against a brick wall. The right
triangle ABE has a hypotenuse of the sum or the radii (R1+R2) and a 
shorter leg of the difference between the radii (R2-R1), but the other 
two angles are unknown and the length of the third side (same as the 
length of the segment between the points of tangency) cannot be 
determined... ouch!

Help?  Thanks!  Tom

        R2-R1
     ____________
     |              /
     |            /
     |          /
     |        /    R2+R1
     |      /
     |    /
     |  /
     |/

Date: 04/05/99 at 22:38:03
From: thammond
Subject: Re: Length of segment of line tangent to two tangent circles 
of different radii

I feel like a total idiot... and Pythagoras would be totally ashamed 
of me. We made this problem up ourselves and then couldn't answer it, 
but... well, I sent a previous note saying your help didn't complete 
the process, but we had to remember that the segment of the tangent 
line could only be put in terms of the radii.  So...

The hypotenuse is the sum of the radii (R1+R2). The one leg of the 
right triangle created by your method is the difference between the 
radii (R2-R1). If we set the other leg, that is, the one congruent to 
the segment we are trying to find, equal to "x", then Pythagoras shows 
us that X equals 2 times the square root of the product of the radii.

X^2 + (R2-R1)^2 = (R1+R2)^2

X =  2*(SQRT)(R1R2)

You perform a wonderful and needed service.  Thank you SO much!
Tom Hammond

Date: 04/06/99 at 12:00:08
From: Doctor Peterson
Subject: Re: Length of segment of line tangent to two tangent circles 
of different radii

Good work, Tom! And you thought this problem up yourselves? Most 
problems I think up myself can't be solved! This is actually a fairly 
standard problem, in part because of the neat answer, as well as the 
neat method.

Yes, Pythagoras is the magic word in a surprising number of problems. 
Did you also notice that the answer is the geometric mean of the 
diameters of the circles? It's surprising how often this turns up in 
geometry, explaining the name "geometric."

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/