MathNet 2026 — Part 05 Question 19

Suppose $a$, $b$, $c$ are complex numbers such that $a + b + c = 0$. Prove that
$$
2(a - b)^2(b - c)^2(c - a)^2 = (a^2 + b^2 + c^2)^3 - 54a^2b^2c^2.
$$