Theory:

Let us look at the graph carefully.
 
Fig_4.svg
 
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, \(y\) \(=\) \(\frac{my_2 + ny_1}{m + n}\).