Problem source

Explain the geometrical relationship between the points in the Argand diagram represented by the complex numbers $z$ and $z\mathrm{e}^{\mathrm{i}\theta}.$ 
Write down necessary and sufficient conditions that the distinct complex numbers $\alpha,\beta$ and $\gamma$ represent the vertices of an equilateral triangle taken in anticlockwise order. 
Show that $\alpha,\beta$ and $\gamma$ represent the vertices of an equilateral triangle (taken in any order) if and only if 
\[
\alpha^{2}+\beta^{2}+\gamma^{2}-\beta\gamma-\gamma\alpha-\alpha\beta=0.
\]
Find necessary and sufficient conditions on the complex coefficients $a,b$ and $c$ for the roots of the equation 
\[
z^{3}+az^{2}+bz+c=0
\]
to lie at the vertices of an equilateral triangle in the Argand digram. 

Solution source

The point $ze^{i\theta}$ is obtained by rotating the point $z$ about $0$ by an angle $\theta$ anticlockwise.

The complex numbers $\alpha, \beta$ and $\gamma$ will form an equilateral triangle iff the angles between each side are $\frac{\pi}{3}$, ie

\begin{align*}
\begin{cases}{\gamma - \beta} &= e^{i \frac{\pi}{3}}({\beta - \alpha}) \\
{\alpha- \gamma} &= e^{i \frac{\pi}{3}}({\gamma- \beta}) \\
{\beta- \alpha} &= e^{i \frac{\pi}{3}}({\alpha- \gamma})\end{cases}
\end{align*}
We don't need all these equations, since the first two are equivalent to the third.

Combining the first two equations, we have

\begin{align*}
&& \frac{\gamma - \beta}{\beta-\alpha} &= \frac{\alpha-\gamma}{\gamma - \beta} \\
\Leftrightarrow && (\gamma - \beta)^2 &= (\alpha-\gamma)(\beta-\alpha) \\
\Leftrightarrow && \gamma^2 +\beta^2 - 2\gamma \beta &= \alpha\beta-\alpha^2-\gamma\beta+\gamma\alpha \\
\Leftrightarrow && \alpha^{2}+\beta^{2}+\gamma^{2}-\beta\gamma-\gamma\alpha-\alpha\beta&=0
\end{align*}
as required.

If the roots of $z^{3}+az^{2}+bz+c=0$ are $\alpha, \beta, \gamma$ then $\alpha+\beta+\gamma = -a$ and $\beta\gamma+\gamma\alpha+\alpha\beta = b$. We also have that $a^2 - 2b = \alpha^2+\beta^2+\gamma^2$. Therefore there roots will lie at the vertices of an equilateral triangle iff $a^2-3b = 0$