I wonder if there is a geometric proof or a short proof of the following:
let $z_1,z_2,z_3$ be three complex numbers of modulus $r$. prove that the number $ \frac{r^4+z_1z_2+z_2z_3+z_3z_1}{z_1+z_2+z_3+z_1z_2z_3} $ is also of modulus $r$.
I wrote everything in trigonometric form and after a page of calculations the result was clear. I wonder if there is a more elegant and shorter proof, maybe using geometry or another approach.