(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 satisfy the conditions

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

all points lie on the unit circle, and thus are of the form .

Without loss of generality I will take .

Thus the remaining condition becomes:

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