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

Since pts are collinear, their slope

is equal

$\Rightarrow \frac{y_3-y_1}{x_3-x_1}=\frac{y_2-y_1}{x_2-x_1}=\frac{y_3-y_2}{x_3-x_2}=k$

$\Rightarrow y_3-y_1=k(x_3-x_1)$

$y_2-y_1=k(x_2-x_1)$

$y_3-y_2=k(x_3-x_2)$

$\Rightarrow \frac{y_1-y_2}{x_1{x}_2}=\frac{k(x_1-x_2)}{x_1{x}_2}=k(\frac{1}{x_2}-\frac{1}{x_1})$ ___ (1)

$\frac{y_2-y_3}{x_2{x}_3}=\frac{k(x_2-x_3)}{x_2{x}_3}=k(\frac{1}{x_3}-\frac{1}{x_2})$ ___ (2)

$\frac{y_3-y_1}{x_3{x}_1}=\frac{k(x_3-x_1)}{x_3{x}_1}=k(\frac{1}{x_1}-\frac{1}{x_3})$ ___ (3)

Adding (1), (2), (3), we have

$\sum \frac{y_1-y_2}{x_1{x}_2}=0$