Question

If the points $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$ are collinear, show that $\sum \left(\frac{y_1-y_2}{x_1{x}_2}\right)=0$, i.e
$\frac{y_1-y_2}{x_1{x}_2}+\frac{y_2-y_3}{x_2{x}_3}+\frac{y_3-y_1}{x_3{x}_1}=0$

Answer 1

Given $(x_1,y_1)(x_2,y_2)(x_3,y_3)$ are collinear

So $\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$

$x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)=0$ ____ (1)

By solving first column

Divide (1) by $x_1{x}_2{x}_3$

$\frac{x_1(y_2-y_3)}{x_1{x}_2{x}_3}+\frac{x_2(y_3-y_1)}{x_1{x}_2{x}_3}+\frac{x_3(y_1-y_2)}{x_1{x}_2{x}_3}=0$

$\frac{y_2-y_3}{x_2{x}_3}+\frac{y_3-y_1}{x_1{x}_3}+\frac{y_1-y_2}{x_1{x}_2}=0$

$\therefore \sum \left(\frac{y_1-y_2}{x_1{x}_2}\right)=0$

$\therefore$ Hence proved.