Complex numbers question - sum of three complex numbers

Question

Seems an easy one but i can't figure it out:

$z_1+z_2+z_3=0$

$|z_1|=|z_2|=|z_3|=1$

Need to prove the following:

$z_1^2+z_2^2+z_3^2=0$

Thanks!

Answer 1

Conjugate $z_1+z_2+z_3=0$ and get $\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}=0$ which simplifies to $z_1z_2+z_1z_3+z_2z_3=0$. Now square the original equation and you are done!

Answer 2

It suffices to prove that

$$z_1+z_2+z_3=0 \iff z_1=e^{i\theta} \quad z_2=we^{i\theta} \quad z_3=w^2e^{i\theta}$$

indicating with $w$ the principal third roots of unity, then

$$z_1^2+z_2^2+z_3^2=e^{2i\theta}+w^2e^{2i\theta}+w^4e^{2i\theta}=e^{2i\theta}(1+w+w^2)=0$$

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