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

Question

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

When it comes to inversion in a circle, I only know two cases:

1. a circle that does not pass through the origin maps onto a circle

2. a circle punctured at the origin maps onto a line that does not pass through the origin.

However, none of these cases mention anything about the radius. Does anyone have an idea?