Vertices of Equilateral Triangle in Complex Plane
Theorem
Let $z_1$, $z_2$ and $z_3$ be complex numbers.
Then:
- $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle
- ${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$
Corollary
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
Sufficient Condition
Let $T$ be the equilateral triangle whose vertices are $z_1$, $z_2$ and $z_3$.
We have that $z_2 - z_1$ and $z_3 - z_1$ are two sides of $T$ which meet at $z_1$.
From the geometry of $T$ it follows that $z_2 - z_1$ is at an angle of $\pi/3$ to $z_3 - z_1$.
Similarly, $z_1 - z_3$ and $z_2 - z_3$ are two sides of $T$ which meet at $z_3$.
From the geometry of $T$ it follows that $z_1 - z_3$ is at an angle of $\pi / 3$ to $z_2 - z_3$.
From Complex Multiplication as Geometrical Transformation/Corollary:
- $(1): \quad z_2 - z_1 = e^{i \pi / 3} \left({z_3 - z_1}\right)$
- $(2): \quad z_1 - z_3 = e^{i \pi / 3} \left({z_2 - z_3}\right)$
Then:
\(\ds \dfrac {z_2 - z_1} {z_1 - z_3}\) | \(=\) | \(\ds \dfrac {z_3 - z_1} {z_2 - z_3}\) | $(1)$ divided by $(2)$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {z_2 - z_1} \paren {z_2 - z_3}\) | \(=\) | \(\ds \paren {z_3 - z_1} \paren {z_1 - z_3}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds {z_2}^2 - z_1 z_2 - z_2 z_3 + z_3 z_1\) | \(=\) | \(\ds - {z_1}^2 - {z_3}^2 + 2 z_3 z_1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds {z_1}^2 + {z_2}^2 + {z_3}^2\) | \(=\) | \(\ds z_1 z_2 + z_2 z_3 + z_3 z_1\) |
$\Box$
Necessary Condition
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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity: Exercise $7$