Why is the ratio of any two sides of an equilateral triangle on a complex plane equal to a complex root of unity?
This question is derived from an excerpt from the following note, which states that:
"If $a_1, a_2, a_3$ are vertices of an equilateral triangle, then triple of complex numbers $a_1 − a_2, a_2 −a_3$ and $a_3 −a_1$ have the same length and the ratio of any consecutive pair is the same (complex) cube root of unity."
I am not sure why the case is true. I know how to represent a ratio of two complex numbers on a complex plane but I am lost on how to prove my question.
Intuitive AND rigorous answer would be greatly appreciated, though either is fine
If $a_1, a_2, a_3$ are the vertices of the triangle then $a_1-a_2, a_2 -a_3, a_3-a_1$ are `vectors' representing the edges of the triangle.
You can turn one edge of an equilateral triangle into the next edge by rotating it by angle $2\pi/3$. This corresponds to multiplication by $e^{2\pi i 3}$, aka the complex cube root of unity.
In other words, $a_1-a_2 = e^{2\pi i /3} (a_2-a_3)$ and so on.