by Sharren M. Thomas
Given two circles and a point on one of the circles. Construct a circle tangent to the two circles with one point of tangency being the designated point.
Click Here for the script tool for construction of the tangent circles.
See below, the solution with included work from my constructions.
Now I will investigate the locus if we have two intersecting circles and we examine the locus of the centers of the constructed tangent circles.
Click here to see a trace. Something interesting I found was that I noticed the trace in this case shows a hyperbola with foci at the original circle centers rather than an ellipse, which is what I found when one was inside the other.
I have to believe, but would definitely have take more cases, but it appears the locus will be either an ellipse or hyperbola.