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 Check out our web site! http://mathforum.org/dr.math/
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