lean_workbook_plus_82357

Let $z_1$ , $z_2$ and $z_3$ be three such complex numbers. We set $a=z_1-z_2$ , $b=z_2-z_3$ and $c=z_3-z_1$ . Obviously, $a+b+c=0$ and the given condition: $z_1^2+z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1$ , is equivalent to $a^2+b^2+c^2=0$ . Hence $\left(a+b+c\right)^2=0=a^2+b^2+c^2+2\left(ab+bc+ca\right)$ so $ab+bc+ca=0$ .