Centre of Invariant Circle under Inversion

Question

Given an inversion of the plane, and a circle invariant under this inversion, what information do we know about the inverse of the centre this circle? (I know that an invariant circle must be orthogonal to the circle of invariant points of the inversion, but this doesn't seem to be very helpful.)

Answer 1

Let $R$ be the inversion radius and $r$ be the radius of the invariant circle. The distance between inversion center and the center of the invariant circle is $\sqrt{R^2+r^2}$. So, it goes to the point on the line that connects two circles with a distance of $R^2/\sqrt{R^2+r^2}$ to the inversion center. By $Euler$'s Theorem, one can show that this point is the projection of intersection points of two circles onto the line through the centers.