Vertices of Equilateral Triangle in Complex Plane/Necessary Condition

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

Let $z_1$, $z_2$ and $z_3$ fulfil the condition:
 * ${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$

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

Proof

 * EquilateralTriangleInComplexPlane.png

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

Then:

Thus $z_2 - z_1$ and $z_3 - z_1$ are at the same angle to each other as $z_1 - z_3$ and $z_2 - z_1$.

Similarly:

Thus $z_2 - z_1$ and $z_3 - z_1$ are at the same angle to each other as $z_1 - z_2$ and $z_3 - z_2$.

Thus all three angles:
 * $\angle z_2 z_1 z_3$
 * $\angle z_1 z_3 z_2$
 * $\angle z_3 z_2 z_1$

are equal.

By definition, therefore, $\triangle z_1 z_2 z_3$ is equilateral.