Problem source

\begin{questionparts}
\item Prove that the equations
$$
\left|z - (1 + \mathrm{i}) \right|^2 = 2
\eqno(*)
$$
and
$$
\qquad \quad \ \left|z - (1 - \mathrm{i}) \right|^2 = 2 \left|z - 1 \right|^2 
$$
describe the same locus in the complex $z$--plane. Sketch this
locus.
\item Prove that the equation
$$
\arg \l {z - 2 \over z} \r = {\pi \over 4} \eqno(**)
$$
describes part of this same locus, and show on your sketch which part.
\item The complex number $w$ is related to $z$ by
\[
w = {2 \over z}\;.
\]
Determine the locus produced in the complex $w$--plane if $z$
satisfies $(*)$. Sketch this locus and
indicate the part of this locus that corresponds to $(**)$.
\end{questionparts}