Unit Vectors in Complex Plane which are Vertices of Equilateral Triangle

Theorem
Let $\epsilon_1, \epsilon_2, \epsilon_3$ be complex numbers embedded in the complex plane such that:


 * $\epsilon_1, \epsilon_2, \epsilon_3$ all have modulus $1$
 * $\epsilon_1 + \epsilon_2 + \epsilon_3 = 0$

Then:


 * $\paren {\dfrac {\epsilon_2} {\epsilon_1} }^3 = \paren {\dfrac {\epsilon_3} {\epsilon_2} }^2 = \paren {\dfrac {\epsilon_1} {\epsilon_3} }^2 = 1$

Proof
We have that:

Thus by Geometrical Interpretation of Complex Subtraction, $\epsilon_1$, $\epsilon_2$ and $\epsilon_3$ form the sides of a triangle.

As the modulus of each of $\epsilon_1$, $\epsilon_2$ and $\epsilon_3$ equals $1$, $\triangle \epsilon_1 \epsilon_2 \epsilon_3$ is equilateral.


 * Equliateral-Triangle-Unit-Sides.png

By Complex Multiplication as Geometrical Transformation:

The same analysis can be done to the other two pairs of sides of $\triangle \epsilon_1 \epsilon_2 \epsilon_3$.

Hence the result.