Let I be an inversion and let p be a circle such that I(p) is also a circle. When do p, I(p) have equal radii?

My attempt: Let I_{C,y} = I and let r be the radius of p, which is fixed, and let y be the radius of I. There are then two cases I broke it down into:



If the center C of I lies outside circle p. Denote the distance between C and the point on p closest to C, D, as x and let E be the antipodal point so that DE is a radius of p and CDE are collinear. Since CD\cdot CI(D) = y^{2} we get CI(D)= \frac{y^{2}}{x} and similarly we shall get CI(E)=\frac{y^{2}}{2r+x} and since I(D)I(E) = CI(E)+CI(D) we want \frac{y^{2}}{x}+\frac{y^{2}}{x+2r} = 2r and similarly, if C is within circle p we get \frac{y^{2}}{x}+\frac{y^{2}}{-x+2r} = 2r

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

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