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.
- $u^2 + v^2 = u v$
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.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity: Exercise $6$