How to solve complex number

Question

how to solve below complex number problem .

The points $A,B,C$ represent the complex numbers $z_1,z_2,z_3$ respectively, and $G$ is the centroid of the triangle $ABC$ . If $4z_1+z_2+z_3=0$, show that the origin is the mid point of $AG$ ?

Answer 1

The centroid G is defined to be $\frac{z_1+z_2+z_3}{3}$. So divide the constraint by 3 to get $(3/3) z_1 + (1/3) (z_1 + z_2 +z_3) = 0$. This implies that $z_1 = -G$. Thus the midpoint of $z_1$ and G must be 0.