MathNet 2016-part-04 — 184 problems (EN–VI) Question 3

Let $a$, $b$, $c \in \mathbb{C}^*$, be distinct complex numbers with equal moduli, such that $a^2 + b^2 + c^2 + ab + ac + bc = 0$. Prove that $a$, $b$, $c$ are the complex coordinates of the vertices of a triangle which is either right angled or equilateral.

Marian Ionescu