Three points $P(h, k)$,$Q(x_{1}, y_{1})$ and $R(x_{2}, y_{2})$ lie on a line. Show that $(h-x_{1})(y_{2}-y_{1}) = (k-y_{1})(x_{2}-x_{1})$.

(N/A) Since points $P$,$Q$,and $R$ are collinear,the slope of line segment $PQ$ must be equal to the slope of line segment $QR$.
The slope of $PQ$ is given by $m_{PQ} = \frac{y_{1}-k}{x_{1}-h}$.
The slope of $QR$ is given by $m_{QR} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$.
Equating the slopes: $\frac{y_{1}-k}{x_{1}-h} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$.
Multiplying both sides by $-1$ in the numerator and denominator of the left side: $\frac{k-y_{1}}{h-x_{1}} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$.
Cross-multiplying gives: $(h-x_{1})(y_{2}-y_{1}) = (k-y_{1})(x_{2}-x_{1})$.
Hence,the points are collinear.