A geometric property of the Möbius transformation (Q6123794)

Summary: 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 \[ f(z_0) = w_0,\quad f(a) = b,\quad z_0 \in \partial D,\quad w_0 \in \partial G,\quad a \in D,\quad b \in G. \]