Question

If z1 z2 and z3 are vertices of an equilateral triangle prove that z1^2+z2^2

Answer 1

as |z1|=|z2|=|z3| so z1,z2,z3 lie on a circle with the edges of triangle subtending an angle of 120 degree.
say arg z2=argz1+120degrees and arg z3=argz1-120degrees
so z1+z2+z3=|z1|cis(arg(z1))+|z2|cis(argz1+120 degrees)+|z3|cis(argz3-120 degrees)=|z1|(0)=0

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