If $z_1,z_2,z_3$  are the vertices of an equilateral triangle with centroid $z_0$ , then $z_1^2+z_2^2+z_3^2$  equals:

$z_0=\frac{z_1+z_2+z_3}{3}$

$\Rightarrow$       $9z_0^2=z_1^2+z_2^2+z_3^2+2\left(z_1{z}_2+z_2{z}_3+z_3{z}_1\right)$

(For equilateral $z_1^2+z_2^2+z_3^2=z_1{z}_2+z_2{z}_3+z_3{z}_1$

$\therefore z_1^2+z_2^2+z_3^2=3z_0^2$ .