Complex Numbers: Triangle and Centroid

Question

Show that $(z_1+z_2+z_3)/3$ is the centroid of a triangle whose vertices are $z_1$, $z_2$, $z_3$. (Hint: The centroid divides the median internally in the ratio of 2:1)

Answer 1

Show that the point $G:=\frac {z_1 + z_2 + z_3}3$ is along the median line through the vertex $z_1$ and the midpoint of the opposite side $\frac {z_2 + z_3}2$ whose parametric equation is $(1-t)z_1+t\frac {z_2 + z_3}2$ with $t\in\mathbb{R}$. The same for the other two medians.

Can you take it from here?

Answer 2

Note that from the hint and following RobertZ suggestion you should arrive to

$$\frac{z_1+z_2+z_3}3=z_1+\frac23\left(\frac{z_2+z_3}2-z_1\right)=z_2+\frac23\left(\frac{z_3+z_1}2-z_2\right)=z_3+\frac23\left(\frac{z_1+z_2}2-z_3\right)$$