Point dividing Line Segment between Two Points in Given Ratio/Proof 2

Proof
Let the coordinates of $A$ be $\tuple {x_1, y_1}$.

Let the coordinates of $B$ be $\tuple {x_2, y_2}$.

Let the coordinates of $R$ be $\tuple {X, Y}$.

Then we have:
 * $\dfrac {x_2 - X} {X - x_1} = \dfrac n m$

and so:
 * $X \paren {m + n} = m x_2 + n x_1$

Similarly for $Y$, giving $R$ as:


 * $\tuple {\dfrac {m x_2 + n x_1} {m + n}, \dfrac {m y_2 + n y_1} {m + n} }$

The result follows.