Let be an inversion and let be a circle such that is also a circle. When do have equal radii?

My attempt: Let and let be the radius of , which is fixed, and let be the radius of . There are then two cases I broke it down into:

If the center of lies outside circle . Denote the distance between and the point on closest to , , as and let be the antipodal point so that is a radius of and are collinear. Since we get and similarly we shall get and since we want and similarly, if is within circle we get

So if the given variables satisfy either equation, depending on where the center of the inversion is, then will have equal radii.

Is this legitimate? Is there an easier way? Any help would be appreciated, thanks.