Complex numbers question - sum of three complex numbers

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!