Vertices of Equilateral Triangle in Complex Plane/Corollary

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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.

if and only if:

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


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.

if and only if:

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


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.

$\blacksquare$


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