Let $\Gamma$ be a circle on the Riemann sphere. Explain what it means to say that two points of the sphere are inverse points for the circle $\Gamma$. Show that, for each point $z$ on the Riemann sphere, there is a unique point $z^{\prime}$ with $z, z^{\prime}$ inverse points. Define inversion in $\Gamma$.

Prove that the composition of an even number of inversions is a Möbius transformation.