\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\un}{\underline}
\parindent=0pt
\begin{document}

{\bf Question}

Let $A$ be the Euclidean circle in ${\bf C}$ with center $2 + i$
and radius $3$.  Construct two (non-equal and) explicit M\"obius
transformations $m$ and $n$ satisfying $m(\overline{\bf R}) =
n(\overline{\bf R}) = A$.

\medskip
\noindent Verify that $m\circ C\circ m^{-1} = n\circ C\circ
n^{-1}$, where $C(z) =\overline{z}$ is complex conjugation.
\medskip

{\bf Answer}

\un{$m(\bar{\bf{R}})=A$}:

Choose 3 points on $A$: $z_1=5+i; z_2=-1+i; z_3=2+4i$

$m^{-1}(z_1)=0, m^{-1}(z_2)=1,m^{-1}(z_3)=\infty$

\begin{eqnarray*} m^{-1}(z) & = & \ds\frac{z-(5+i)}{z-(2+4i)}
\cdot \ds\frac{(-1+i)-(2+4i)}{(-1+i)-(5+i)}\\ & = &
\ds\frac{z-(5+i)}{z-(2+4i)} \cdot \ds\frac{(-3-3i)}{-6+0}\\ & = &
\ds\frac{z-(5+i)}{z-(2+4i)} \cdot \ds\frac{(1+i)}{2}\\ & = &
\ds\frac{(1+i)z-4-6i}{2z-4-8i}\\ m(z) & = &
\ds\frac{(-4-8i)z+4+6i}{-2z+1+i} \end{eqnarray*}

Choose another three points on $A$: $w_1=-1+i; w_2=2+4i; w_3=2-2i$

$n^{-1}(w_1)=0; n^{-1}(w_2)=1; n^{-1}(w_3)=\infty$

\begin{eqnarray*} n^{-1}(z) & = & \ds\frac{z-(-1+i)}{z-(2-2i)}
\cdot \ds\frac{(2+4i)-(2-2i)}{(2+4i)-(-1+i)}\\ & = &
\ds\frac{z+1-i}{z-2+2i} \cdot \ds\frac{(6i)}{3+3i}\\ & = &
\ds\frac{z+1-i}{z-2+2i} \cdot \ds\frac{(2i)}{1+i}\\ & = &
\ds\frac{2iz+2+2i}{(1+i)z-4}\\ n(z) & = &
\ds\frac{-4z-2-2i}{-(1+i)z+2i}
\end{eqnarray*}

\newpage
\begin{eqnarray*}
m \circ C \circ m^{-1}(z) & = & m(\overline{m^{-1}(z)})\\ & = &
m\left[\ds\frac{(1-i)\bar z-4+6i}{2\bar z-4+8i}\right]\\ & = &
\ds\frac{(-4-8i)[(1-i)\bar z-4+6i]+(4+6i)(2 \bar
z-4+8i)}{-2[(1-i)\bar z-4+6i]+(1+i)(2\bar z-4+8i)}\\ & = &
\ds\frac{(-4+8i)\bar z+16i}{4i\bar z-4-8i}
\end{eqnarray*}

\begin{eqnarray*}
n \circ C \circ n^{-1}(z) & = & n(\overline{n^{-1}(z)})\\ & = &
n\left[\ds\frac{-2i\bar z+2-2i}{[(1-i)\bar z-4]}\right]\\ & = &
\ds\frac{-4(-2i\bar z+2-2i)-(2+2i)[(1-i)\bar z-4]}{-(1+i)(-2i\bar
z+2-2i)+2i[(1-i)\bar z-4]}\\ & = & \ds\frac{(-4+8i)\bar
z+16i}{4i\bar z-4-8i} = m \circ C \circ m^{-1}(z),\ \rm{as\
desired}
\end{eqnarray*}
\end{document}
