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## Homework Statement

Let ##z_1,z_2,z_3## be three complex numbers that lie on the unit circle in the complex plane, and ##z_1+z_2+z_3=0##. Show that the numbers are located at the vertices of an equilateral triangle.

## Homework Equations

## The Attempt at a Solution

As far as I understand, I need to show that ##|z_1-z_2|=|z_1-z_3|=|z_2-z_3|##. I have tried a lot of things but no where near a solution. I know that ##|z_i|=1## ,(i=1,2,3), since it's the distance of the number from the origin. Also tried to write ##z_i## in terms of the other two(it gave me that the distance from ##z_i## to ##(-z_j)## is 1) , but still nothing. I am missing something and not sure what.

__Please, don't give me the answer, just a direction of thinking.__

Thank you.