First: in general, it is not solvable. There are points which cannot
lie on a circle tangent to two others. E.g., the point cannot be inside
either circle. And there are 4 other, "small", areas that are difficult
to describe, but are easily shown on a graphic (later).

Second, and more importantly, the radius of the solution circle can
approach infinity. So, in practical terms, it becomes unsolvable at
some point, depending upon whether you are using CAD or a drawing board G.

Now, there are three cases for the solution: where the given circles are
externally or internally tangent to the solution circle, or one of
each. Do your problems always fall into one of these classes? Also, if
the given circles can overlap, it may be a different set of solution
classes (I haven't thought that through). Do your circles ever overlap?

I love plane geometry,
Bob