A geometric property of the Möbius transformation

Abstract

We show that, for any triple $(z_0,a,a^{\star })$ of distinct points in $\mathbb{C}$ there exists a Möbius circle $C$ such that $z_0\in C$, and $a$ and $a^{\star }$ are conjugate with respect to $C$. We use this fact to avoid tedious calculations when constructing a Möbius transformation $f:D\to G$, where $D$ and $G$ are Möbius disks, and $f$ satisfies the conditions