Question

if $z_1,z_2,z_3$ are non - zero complex such that $z_1+z_2+z_3=0$ and $z_1^{-1}+z_2^{-1}+z_3^{-1}=0$ then prove that the given points are the verices of an equilateral triangle. Also show that $|z_1|=|z_2|=|z_3|$

Answer 1

Given $z_1+z_2+z_3=0$, and form second relation $z_2{z}_3+z_3{z}_1+z_1{z}_2=0$

$\therefore z_2{z}_3=z_1(z_2+z_3)=z_1(-z_1)=z_1^2$

$\therefore z_1^3=z_1{z}_2{z}_3=z_1^3=z_3^3$

$|z_1{|}^3=|z_2{|}^3=|z_3{|}^3$

$\Rightarrow |z_1|=|z_2|=|z_3|$