### Theory:

Let us look at the graph carefully.

From the graph, the line $$AB$$ is divided at $$P$$ in the ratio $$m : n$$.

Therefore, $$\frac{AP}{AB} = \frac{m}{n}$$.

So, $$A'P' : P'B'$$ is also $$m : n$$.

$$\frac{A'P'}{A'B'} = \frac{m}{n}$$

$$n(A'P') =$$ $$m(A'B')$$

$$n(x - x_1) =$$ $$m(x_2 - x)$$

$$nx - nx_1 = mx_2 - mx$$

$$mx + nx = mx_2 + nx_1$$

$$x$$ $$=$$ $$\frac{mx_2 + nx_1}{m + n}$$

Similarly, $$y$$ $$=$$ $$\frac{my_2 + ny_1}{m + n}$$.