Question

The three points $z_1,z_2,z_3$ are connected by the relation ${az}_1+{bz}_2+{cz}_3=0$, where $a+b+c=0$. Prove that three points are collinear.

Answer 1

$z_1=(x_1+i{y}_1)=(x_1,y_1)$
$a{z}_1+b{z}_2+c{z}_3=(a{x}_1+b{x}_2+c{x}_3)+i(a{y}_1+b{y}_2+c{y}_3)=0$
Above relation implies that
$a{x}_1+b{x}_2+c{x}_3=0$
$a{y}_1+b{y}_2+c{y}_3=0$
$a+b+c=0$ (given)
Eliminating $a,b,c$ we get
$\left|\begin{array}{ccc}{x}_1& x_2& x_3\\ y_1& y_2& y_3\\ 1& 2& 3\end{array}\right|=0$ or $\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$
$\therefore$ $\Delta =0$ i.e., the points are collinear.