Problem source

The distinct points $P_{1},P_{2},P_{3},Q_{1},Q_{2}$ and $Q_{3}$
in the Argand diagram are represented by the complex numbers $z_{1},z_{2},z_{3},w_{1},w_{2}$
and $w_{3}$ respectively. Show that the triangles $P_{1}P_{2}P_{3}$
and $Q_{1}Q_{2}Q_{3}$ are similar, with $P_{i}$ corresponding to
$Q_{i}$ ($i=1,2,3$) and the rotation from $1$ to $2$ to $3$ being
in the same sense for both triangles, if and only if 
\[
\frac{z_{1}-z_{2}}{z_{2}-z_{3}}=\frac{w_{1}-w_{2}}{w_{1}-w_{3}}.
\]
Verify that this condition may be written 
\[
\det\begin{pmatrix}z_{1} & z_{2} & z_{3}\\
w_{1} & w_{2} & w_{3}\\
1 & 1 & 1
\end{pmatrix}=0.
\]
\begin{questionparts}
\item Show that if $w_{i}=z_{i}^{2}$ ($i=1,2,3$) then triangle
$P_{1}P_{2}P_{3}$ is not similar to triangle $Q_{1}Q_{2}Q_{3}.$ 
\item Show that if $w_{i}=z_{i}^{3}$ ($i=1,2,3$) then triangle
$P_{1}P_{2}P_{3}$ is similar to triangle $Q_{1}Q_{2}Q_{3}$ if and
only if the centroid of triangle $P_{1}P_{2}P_{3}$ is the origin. 
{[}The \textit{centroid }of triangle $P_{1}P_{2}P_{3}$ is represented
by the complex number $\frac{1}{3}(z_{1}+z_{2}+z_{3})$.{]} 
\item Show that the triangle $P_{1}P_{2}P_{3}$ is equilateral if
and only if 
\[
z_{2}z_{3}+z_{3}z_{1}+z_{1}z_{2}=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}.
\]
\end{questionparts}