Question

Three points $P(h,k),Q(x_1,y_1)\text{and}R(x_2,y_2)$ lie on a line. Show that $(h-x_1)(y_2-y_1)=(k-y_1)(x_2-x_1)$

Answer 1

Given: $P\equiv (h,k),Q\equiv (x_1,y_1),R\equiv (x_2,y_2)$

Slope of line $PQ=\frac{y_1-k}{x_1-h}$
Slope of line $QR=\frac{y_2-y_1}{x_2-x_1}$

Since the points $P,Q\text{and}R$ are collinear.
So, $\text{Slope of}PQ=\text{Slope of}QR$
$\Rightarrow \frac{y_1-k}{x_1-h}=\frac{y_2-y_1}{x_2-x_1}$
$\Rightarrow \frac{k-y_1}{h-x_1}=\frac{y_2-y_1}{x_2-x_1}$
$\Rightarrow (h-x_1)(y_2-y_1)=(k-y_1)(x_2-x_1)$

Hence proved.​​​​