If the complex numbers z1,z2,z3 represent the the vertices of an equilateral triangle such that |z1|=|z2|=|z3| , then prove that z1+z2+z3=0
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Solution
Let z1,z2,z3 represent the vertices A1,A2,A3 of ΔA1A2A3 respectively and let O be the origin. Then z1=−−→OA1,z2=−−→OA2,z3=−−→OA3 Since |z1|3=|z2|3=|z3|3 we have OA1=OA2=OA3 This show that O is the circum centre of the ΔA1A2A3 . ∠A1OA2=∠A2OA3=∠A3OA1=2π3 Hence we can write z2=z1e2πi/3=z1ωandz3=z1e4πi/3=z3ω2 Hence z1+z2+z3=z1(1+ω+ω2)=0