I did assume that the two radiuses were tangent to the other circles so that may be incorrect.
For simplicity i used the circle tan tan radius, but there is another way if you want a brute force method.
If you assume any two circles are tangent to one another at a distance of Rt around the center circle 1.
Rt = r1 + r2
So basically around your first circle you draw an additional circle that is a radius of Rt.
so if we need to find the point at which three circles are connected only by tangents you just repeat the last step with the new circle.
Rt2 = r3 + r2 (assuming r2 is the circle common to both tangents)
If they can be tangent to each other there will be a location where Rt and Rt2 intersect. This is the location where you put the center of r2.

CADnoob
