Let complex numbers $z_1$, $z_2$, and $z_3$ satisfy $|z_1|=|z_2|=|z_3| = r > 0$. If $z_1+z_2+z_3 \ne 0$, prove $$\frac{z_1z_2+z_2z_3+z_3z_1}{z_1+z_2+z_3}=r$$