Quandaries
and Queries 

I have a mathematical problem I am trying to solve that involves some high level geometry/trig. I know you typically service high school levels and below. However, I am searching for guidance where ever I can. I have a horizontal line (that is treated as a datum line or the X axis), with two circles having their center points at different heights from that line (X1,Y1 & X2,Y2). The two circles are also at different diameters (R1 & R2). Both circles and the line (XAxis) do not intersect nor are they tangent. My goal is to determine the maximum diameter of an inscribed circle that will fit between all three. I can easily use autocad to find this value, but I need to express it as an equation. I have researched some info on Apollonius but nothing that I can clearly apply or explains the contruction of the three tangent points to both circles and the line to construct the inscribed circle. Attached is a sketch of the problem. The XY Origin was not define, but can be established at any place. Thanks for any help, Keith 

Hi Keith, I redrew your diagram and changed the notation so that
one of the circles has its centre on the Yaxis. As in your description,
the horizontal line is the Xaxis. Consider the two circles in the diagram below. Since the distance between the centres is r + y, Applying the same argument to the pair of circle below, Thus the radius, y, of the circle with centre (x,y) can be found by solving 

