SOLUTION: prove that if z1 z2 z3 are complex numbers on the unit circle such that z1+z2+z3=0, then z1 z2 z3 are vertices of an equilateral triangle

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: prove that if z1 z2 z3 are complex numbers on the unit circle such that z1+z2+z3=0, then z1 z2 z3 are vertices of an equilateral triangle      Log On


   

Question 1104828: prove that if z1 z2 z3 are complex numbers on the unit circle such that z1+z2+z3=0, then z1 z2 z3 are vertices of an equilateral triangle
Answer by greenestamps(13200) About Me  (Show Source):
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Let z1 = a+bi; z2 = c+di; z3 = e+fi.

Since z1+z2+z3=0, a+c+e=0 and b+d+f=0.

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.

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.

But we know the three points are the same distance from the centroid, because they are points on the unit circle.

That means the three medians of the triangle are the same length.

And that means the three sides of the triangle are the same length.

So the three points are the vertices of an equilateral triangle.