Tangency Point of Two Circles: What is this about?
A Mathematical Droodle

Assume two circles tangent to AB at A and B are also tangent to each other. What is the locus of their common point T when their radii change?

The locus is a circle with center at the midpoint of AB and the radius half its length. To see that draw the common tangent to the two circles at T to its intersection with AB at O. Then, since two tangents from a point to a circle are equal,