(I've been taking the z's to be points in the Argand plain, but I suppose you could simply think of the z's as (x, y) points.)Four points $\displaystyle z_1, z_2, z_3, z_4$ satisfy the conditions

$\displaystyle z_1 + z_2 + z_3 + z_4 = 0$

$\displaystyle |z_1| = |z_2| = |z_3| = |z_4| = 1$

Show that the points lie either at the vertices of a square inscribed in the unit circle or else coincide in pairs.

I was able to solve a similar problem involving 3 points and an inscribed equilateral triangle, but I've only been able to go so far with the 4 point problem.

Here's what I've got so far, and you all can redirect me or give me clue how to continue as you will.

Since

$\displaystyle |z_1| = |z_2| = |z_3| = |z_4| = 1$

all points lie on the unit circle, and thus are of the form $\displaystyle z_n = e^{i \theta_n}$.

Without loss of generality I will take $\displaystyle z_1 = 1$.

Thus the remaining condition becomes:

$\displaystyle e^{i \theta _2} + e^{i \theta _3} + e^{i \theta _4} = -1$

I'm rather stuck at this point. I used a similar approach to the three point problem and it worked nicely. Obviously it isn't working here because I've got that extra variable playing around. Is there, perhaps, a different approach that would work for both problems, or is there another simplifying step I'm missing for the 4 point problem?

-Dan