$z_1,z_2,z_3 \in \mathbb{C}$ Prove that $z_1^2+z_2^2+z_3^2=z_1z_2+z_1z_3+z_2z_3$ is true iff $z_1,z_2,z_3$ are points of a equilateral triangle.

Question

$z_1,z_2,z_3 \in \mathbb{C}$ Prove that $z_1^2+z_2^2+z_3^2=z_1z_2+z_1z_3+z_2z_3$ is true iff $z_1,z_2,z_3$ are points of a equilateral triangle.(not necessarily in the coordinate begining)

Ive proved for example that $|z_1|=|z_2|=|z_3|$ and $z_1+z_2+z_3$ that mean that these numbers are points of a equilateral triangle in the coordinate begining, using some trig identities, but this one I really don't know how to go about doing.