Question

Let $z_1,z_2$ and $z_3$ be three complex numbers and $a,b,c\in R$ such that $a+b+c=0$ and $a{z}_1+b{z}_2+c{z}_3=0$ then show that $z_1,z_2$ and $z_3$ are collinear.

Answer 1

Given:$a+b+c=0$ .....$\left(1\right)$

and $a{z}_1+b{z}_2+c{z}_3=0$ .....$\left(2\right)$

$\Rightarrow a{z}_1+b{z}_2-(a+b)z_3=0$ from eqn$\left(1\right)$

or $z_3=\frac{a{z}_1+b{z}_2}{a+b}$

It follows that $z_3$ divides the line segment joining $z_1$ and $z_2$ internally in the ratio $b:a$.

Hence $z_1,z_2$ and $z_3$ are collinear.