Vertices of Equilateral Triangle in Complex Plane/Necessary Condition

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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:

\(\ds {z_1}^2 + {z_2}^2 + {z_3}^2\) \(=\) \(\ds z_1 z_2 + z_2 z_3 + z_3 z_1\)
\(\ds \leadsto \ \ \) \(\ds {z_2}^2 - z_1 z_2 - z_2 z_3 + z_3 z_1\) \(=\) \(\ds - {z_1}^2 - {z_3}^2 + 2 z_3 z_1\)
\(\ds \leadsto \ \ \) \(\ds \paren {z_2 - z_1} \paren {z_2 - z_3}\) \(=\) \(\ds \paren {z_3 - z_1} \paren {z_1 - z_3}\)
\(\ds \leadsto \ \ \) \(\ds \dfrac {z_2 - z_1} {z_1 - z_3}\) \(=\) \(\ds \dfrac {z_3 - z_1} {z_2 - z_3}\)

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:

\(\ds {z_1}^2 + {z_2}^2 + {z_3}^2\) \(=\) \(\ds z_1 z_2 + z_2 z_3 + z_3 z_1\)
\(\ds \leadsto \ \ \) \(\ds - {z_2}^2 - {z_1}^2 + 2 z_1 z_2\) \(=\) \(\ds {z_3}^2 - z_1 z_3 - z_2 z_3 + z_1 z_2\)
\(\ds \leadsto \ \ \) \(\ds \paren {z_2 - z_1} \paren {z_1 - z_2}\) \(=\) \(\ds \paren {z_3 - z_1} \paren {z_3 - z_2}\)
\(\ds \leadsto \ \ \) \(\ds \dfrac {z_2 - z_1} {z_3 - z_2}\) \(=\) \(\ds \dfrac {z_3 - z_1} {z_1 - z_2}\)

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.

$\blacksquare$


Sources