Complex numbers question - sum of three complex numbers 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!

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


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*showing certain vertices form an equilateral triangle
