Sending the point where all three outer circles meet to infinity you get two parallel lines (they only touch at infinity) and one orthogonal line since inversion preserves angles and it's clear that the two semicircles are orthogonal. Then you are left with two circles that meet all three lines and it becomes easy to figure out their exact position and radius.

Sadly that is where my knowledge of inversion stops, but if I had to I suppose I could reconstruct the equations from first principles.

If you know that the lines must go through the small circle's center, it becomes a fairly simple geometrical problem with three Pythagorean equations and three variables (x, y, r).