Vertices of Equilateral Triangle in Complex Plane/Corollary

Corollary to Vertices of Equilateral Triangle in Complex Plane
Let $u, v \in \C$ be complex numbers.

Then:
 * $0$, $u$ and $v$ represent on the complex plane the vertices of an equilateral triangle.


 * $u^2 + v^2 = u v$
 * $u^2 + v^2 = u v$

Proof
From Vertices of Equilateral Triangle in Complex Plane:


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




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

Setting $z_1 \gets u$, $z_2 \gets v$, $z_3 \gets 0$ we have:


 * $u^2 + v^2 + 0^2 = u v + v \times 0 + 0 \times u$

and the result follows.