How find this value $\left|\frac{z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3}}{z_{1}+z_{2}+z_{3}}\right|$

Question

let three complex $z_{1},z_{2},z_{3}$ such $$z_{1}+z_{2}+z_{3}\neq 0,|z_{1}|=|z_{2}|=|z_{3}|=1$$ Find this value $$\left|\dfrac{z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3}}{z_{1}+z_{2}+z_{3}}\right|$$

My idea:if $z_{1},z_{2},z_{3}$ is real numbers,and such $z_{1}=z_{2}=z_{3}=1$,then we easy to find this value $$\left|\dfrac{z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3}}{z_{1}+z_{2}+z_{3}}\right|=1$$ But other complex case,I can't,Thank you

Answer 1

Note that $$\lvert z_2z_3+z_3z_1+z_1z_2\rvert=\lvert z_1z_2z_3\rvert\cdot\lvert z_1^{-1}+z_2^{-1}+z_3^{-1}\rvert=\lvert\overline z_1+\overline z_2+\overline z_3\rvert=\lvert z_1+z_2+z_3\rvert$$ Can you point out the reason of each equality?