For those formulas I use “Geometry Expressions” and “Maple”.
I use those software programs as a hobbyist, just like I use Rhino for my hobby.
I have to say that “Geometry Expressions” is very outdated. (for example 32-bit). It has quirks and crashes often. On the other hand, the program is unique, there is no comparable software. It is based on symbolic calculations and constraints. “Geogebra” for example is not based on constraints and cannot calculate the presented geometry as symbols. Furthermore, “Geogebra” is of course great and can do many things much better than “Geometry Expressions”.
I also use “Maple” to simplify the formulas. It probably won’t be simpler than the formulas I used here.
The formulas used here are for this specific situation. It may even be necessary to swap circle A and B, for the following reason: some formulas work with triangles (circle centers) A B C (clockwise) and some with triangle A C B (counterclockwise).
I am thinking about a general Grasshopper definition for tantantan with logic. In the example above that would be: “the tangent circle is around circle A and B, and is outside circle C”. It is difficult because circles can also overlap. The tangent circle can therefore be in, around or outside A, B or C. If the tangent circle is in A and B, A and B overlap. It will take a while before I have made it.
But in any case; nice that I could help you.
@Tom_P , Your illustration shows that there are many possibilities. And this does not even show the overlapping situations.