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



Need to prove the following:




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!

  • 2
    $\begingroup$ +1, simple and elegant. $\endgroup$ – rogerl Oct 31 '19 at 21:34
  • $\begingroup$ great! thanks! :) $\endgroup$ – user3343396 Oct 31 '19 at 21:36
  • $\begingroup$ you are welcome and @rogerl thank you for the comment $\endgroup$ – Conrad Oct 31 '19 at 21:37
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    $\begingroup$ I agree really nice solution! $\endgroup$ – user Oct 31 '19 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


Refer to the related


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