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Formula for Common Tangents

Date: 5/27/96 at 23:57:58
From: Donn Ristau
Subject: Tangents

Dr. Math,

I am looking for a simple formula or proof that will define the common 
tangent of two circles of different diameters, given that the center 
of each circle lies on the X axis of a XY graph. The X co-ordinates of 
the intersection of each circle with the X axis are known.  I want to 
be able to calculate the X and Y values of the point on each circle 
that touches the line that is tangent to both circles.  The formula 
should work for circles that intersect as well as circles that do not. 

Thanks for any help you can give.

Donn Ristau

Date: 5/28/96 at 17:21:25
From: Doctor Ken
Subject: Re: Tangents

Hello -

Here are some hints on how you can find the formula you're looking 
for. I've made a picture:


This picture is copied from a compass-and-straight edge construction I 
did of the problem. In this message, I'll use the labels that are in 
that picture.

Let's say that if there is one circle that's smaller than the other, 
it's the circle on the right (the case where it's not is really 
similar). Then if the big circle has radius R, and the little circle 
has radius r, then OC has length R-r. Now look at triangle OCD. It's a 
right triangle, and by using the Pythagorean Theorem, we can find out 
what the length CD is (since OD is also given).  

So now we know everything about triangle COD, including the slope of 
the line through CD. Notice that this is the same as the slope of the 
common tangent line you're looking for. So if the equation for the 
tangent line you're looking for is y = mx + b, you've found m. I'll 
leave it up to you to find b. Good luck, and write back if you're 
still confused!

-Doctor Ken,  The Math Forum
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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