Vertices of Equilateral Triangle in Complex Plane

Theorem
Let $z_1$, $z_2$ and $z_3$ be complex numbers.

Then:
 * $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle




 * ${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$

Proof

 * EquilateralTriangleInComplexPlane.png