Inversion in a circle

Summary

Inversion in a circle with center $O$ and radius $r$ sends a point $P \ne O$ to a point $P'$ on the ray from $O$ through $P$. The point $P'$ is chosen so that $\abs{OP} \cdot \abs{OP'} = r^2$. It swaps the inside and the outside of the circle and fixes the circle pointwise. It maps circles and lines to circles and lines, and it preserves angles. This makes it a basic tool of inversive geometry and of geometric constructions.