# 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!

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!

• +1, simple and elegant. – rogerl Oct 31 at 21:34
• great! thanks! :) – user3343396 Oct 31 at 21:36
• you are welcome and @rogerl thank you for the comment – Conrad Oct 31 at 21:37
• I agree really nice solution! – user Oct 31 at 21:39

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