How does this equality on vertices in the complex plane imply they are vertices of an equilateral triangle?

Solution 1:

The sides $(a_1,a_2)$ and $(a_1,a_3)$ have the same length (because $|x|=1$) and their nonoriented angle is $\pi/3$ (because $\arg(x)=\pm\pi/3$). Since the triangle $a_1a_2a_3$ is isosceles at vertex $a_1$, its angles at vertices $a_2$ and $a_3$ are equal. The angle at $a_1$ is $\pi/3$ and the sum of the three angles is $\pi$. Hence the angle at each vertex is $\pi/3$. QED.

Solution 2:

You're very close to the solution. Remember that you shifted everything to the origin, so that $a_2 - a_1$ is the location of the second vertice and $a_3 - a_1$ the location of the third vertice. Your relation says that $$ a_2 - a_1 = e^{\pm i \pi /3} (a_3 - a_1) $$ hence they are at the same distance of your origin (since if you compute modulus in these cases they are the same on both sides) and the angle between them is $\pi/3$, which means the triangle is equilateral (by doing your work with $a_2$ and $a_3$ you actually get the 2 other angles).

In other words, you did all the work, you only needed to interpret what you've found correctly.

Hope that helps,

Solution 3:

If $\frac{a_2 - a_1}{a_3 - a_1} = e^{\pm i \pi/3}$ we have $|a_2 - a_1| = |a_3 - a_1|$. Moreover, $a_3 - a_2 = (a_3 - a_1) + {a_1 - a_2} = (1 - e^{\pm i \pi/3}) (a_3 - a_1)$. But $1 - e^{\pm i \pi/3} = 1/2 \mp i \sqrt{3}/2 = e^{\mp i \pi/3}$ so also $|a_3 - a_2| = |a_3 - a_1|$. Thus all three sides of the triangle have the same length.