Answer
Please see proof in the work step by step section below.
Work Step by Step
Let $(x_{1},y_{1})$ and $(x_{2},y_{2})$ lie on a line L with the general equation y=mx+c where m is the gradient and c is the y-intercept.
The gradient of L is therefore = $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$.
If $(x^*_{1},y^*_{1})$ and $(x^*_{2},y^*_{2})$ lie on Line L also, they would also satisfy the same equation y=mx+c.
Hence, the gradient based on these points, $\frac{y^*_{2}-y^*_{1}}{x^*_{2}-x^*_{1}}$ , should be equal to the gradient m of line L.
∴m=$\frac{y^*_{2}-y^*_{1}}{x^*_{2}-x^*_{1}}$
Substituting m=$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$, we get:
$\frac{y^*_{2}-y^*_{1}}{x^*_{2}-x^*_{1}}$ =$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ (shown)
