Show that $\frac1x+\frac1y+\frac1z=0$ 
We know that $x=\operatorname{cis}\alpha$, $y=\operatorname{cis}\beta$, $z=\operatorname{cis}\gamma$ and that $x+y+z=0$. Prove that $\frac1x+\frac1y+\frac1z=0$.

The only thing I could get out of this is that $\frac1x+\frac1y+\frac1z=\frac{x+y+z}{xyz}$, and the numerator equals 0 but to be honest I'd really like to know how to prove this properly.

Hint: $x=\operatorname{cis}\alpha\iff|x|=1\iff\bar x=\frac1x$, and $x+y+z=0\iff\bar x+\bar y+\bar z=0$.
