If $a,b,c$  are three real numbers such that $a+b+c=0$ (at least one $a,b,c$ is not equal to zero) and $a{z}_1+b{z}_2+c{z}_3=0,$ then $z_1,z_2,z_3$ :

$a{z}_1+b{z}_2+c{z}_3=0,$ $a+b+c=0.$  at least one of $a,b,c\ne 0$ and $a,b,c\in R$ . Let $a\ne 0$

Then $z_1=\frac{b{z}_2+c{z}_3}{b+c}$                      $\left[\because -a=b+c\right]$

Thus, $z_1$ divides the line segment joining $z_2$  and $z_3$ in the ratio$c:b$ and therefore $z_1,z_2,z_3$ are collinear