<s>[INST] Geometric images of complex numbers $z$ such that triangle with vertices $z, z^2,z^3$ is a right angled triangle. 
Find the geometric images of complex numbers $z$ such that triangle with vertices $z, z^2,z^3$ is a right angled triangle.

My try:
Let the complex number $z=x+yi$
Hence we need to find the locus of the point $(x,y)$ in the argand plane such that the points $(x,y)$, $(x^2-y^2, 2xy)$, $(x^3-3y^2x, 3x^2y-y^3)$
Now here I could apply the formula that the product of the gradient of two sides of triangle formed by these points is $-1$ but the method is a bit cumbersome.  Is there any other geometrical interpretation of it or any algebraic simplification further?
 [/INST] Let the squared-sides of the triangle be
$$\begin{align}
a^2 &\;:=\; |z^1-z^2|^2 \;=\; |z|^2 |1-z|^2 \\
b^2 &\;:=\; |z^2-z^3|^2 \;=\; |z|^4 |1-z|^2 \\
c^2 &\;:=\; |z^3-z^1|^2 \;=\; |z|^2 |1-z|^2 |1+z|^2
\end{align}$$
Trivially, we have a Pythagorean triple if $z=0$ or $z=1$. Otherwise, we can divide each of $a^2$, $b^2$, $c^2$ by $|z|^2 |1-z|^2$ to get, for $z = r(\cos\theta + i \sin\theta)$ with $r\neq0$ and $\theta\neq 0$,
$$a_\star^2 = 1 \qquad b_\star^2 = |z|^2 = r^2 \qquad c_\star^2 = |1+z|^2 =1+r^2+2r\cos\theta$$
so that $(a, b, c)$ is a (non-trivial) Pythagorean triple if and only if $(a_\star, b_\star, c_\star)$ is. Now, let's consider cases:

$$c_\star^2 = a_\star^2 + b_\star^2 \implies 1+r^2+2r\cos\theta = 1 + r^2 \implies \cos\theta = 0 \implies z = \pm p i \quad(p>0)$$

The diagram shows $z_1 = pi$ and $z_2 = -q i$. One may recognize that this is effectively the geometric mean configuration, which makes sense, since $|z_i^2|^2 = |z_i||z_i^3|$.


$$\begin{align}
a_\star^2 &= b_\star^2 + c_\star^2 \implies 1=r^2+1+r^2+2r\cos\theta \implies 2r\left(r+\phantom{r}\cos\theta\right) = 0 \implies r = -\cos\theta \\
b_\star^2 &= a_\star^2 + c_\star^2 \implies r^2 = 1 + 1 + r^2+2 r \cos\theta \implies 2\;\left(1+r\cos\theta\right) = 0 \implies r = -\sec\theta
\end{align}$$

Since $r$ must be non-negative, we restrict $z$ to Quadrants II and III. The diagram below shows typical solutions $z_3$ and $z_4$.

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