Distance between Points of Tangency

Date: 02/16/2003 at 19:20:37
From: Glavy
Subject: Geometry

Two circles, one of radius 5, the other of radius 8, intersect at 
exactly one point, and the center of each circle lies outside the 
other circle. A line is externally tangent to both circles. Find the 
distance between the two points of tangency. 

Thank you!

Date: 02/17/2003 at 03:49:18
From: Doctor Floor
Subject: Re: Geometry

Hi, Glavy,

Thanks for your question.

Let C1 and C2 be the centers of the two circles, where C1 belongs to 
the circle with radius 5. Let T1 and T2 be the points of tangency of 
these circles with the common externally tangent line. 

We know that C1T1 = 5 and C2T2 = 8 and that both are perpendicular to 
the common tangent T1T2. We also know that C1C2 = 5+8 = 13.

Now we have the following sketch:


   T1------ ? --------T2
    |                 |
   5|                 |5
    |                 | 
    |                 |
    |                 |
   C1-----------------X
      `-._            |
           `-._       |3
           13   `-._  |
                      C2

In this sketch C1X is parallel to T1T2 and thus perpendicular to 
T2C2. Now we can find C1X and thus T1T2 by Pythagoras' Theorem.

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/