Question 1104828
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Let z1 = a+bi; z2 = c+di; z3 = e+fi.<br>
Since z1+z2+z3=0, a+c+e=0 and b+d+f=0.<br>
The point ((a+c+e)/3,(b+d+f)/3) is the centroid of the triangle -- the intersection of the medians.  So (0,0) is the centroid of the triangle.<br>
The medians of a triangle intersect at a point that divides each median into two segments with one segment twice the length of the other.<br>
But we know the three points are the same distance from the centroid, because they are points on the unit circle.<br>
That means the three medians of the triangle are the same length.<br>
And that means the three sides of the triangle are the same length.<br>
So the three points are the vertices of an equilateral triangle.