How to find the complex numbers?

Question

to determine all non-zero z complex numbers, for which a ^ 2 * z, a*z ^ 2 and z ^ 3 are the apices of the vertices of an equilateral triangle.

Answer 1

Solve $|a^2z-az^2|=|z^3-az^2|=|z^3-a^2z|,$ representing the sides of the triangle. Since the sides have a non-zero lenght, we get $|a|=|z|=|z+a|.$ Therefore a diagonal of the rhombus with sides of the lenght $|a|$ has the same lenght and cuts the rombus in two equilateral triangles. (Imagine three vectors in the complex plane, originating in $0.$) For given $a=|a|e^{i\alpha},\; a\neq 0$ we get $z=|a|e^{i(\alpha\pm 2\pi/3)}.$