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### 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

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

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

\
\    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

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

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

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/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Triangles and Other Polygons

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