Subject: Re: Geometry question
Date: Tue, 15 Apr 1997 16:44:06 -0400
From: Alexander Bogomolny
If I understood your question correctly, yours is a particular case of a more general problem:
Find a circle tangent to two intersecting lines and a circle.
Now, first of all, a solution:
Draw lines parallel to the given ones but also tangent to the given circle. There will be several pairs of such lines. Pick any two. In thus obtained parallelogram draw the diagonal emanating from the point M of intersection of the two original lines. This diagonal MM' intersects the given circle at a point P. Let R' be the radius of the given circle. Then the raduis R of the circle you are after satifsies R/R' = MP/PM'. The line through the centers of the two circles passes through P.
The theory behind the solution is based on the observation that the point where two circles touch is the center of a similarity transformation which in this case is the product (successive execution) of two simpler ones: reflection in a point and homothety. Homothety is a pure similarity: expand (or contract) distances from a center with a fixed coefficient.
Similarity transformations map lines into lines and parallel lines into parallel lines.