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

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Let $A$ and $B$ be points whose position vectors relative to an origin $O$ of a Euclidean space are $\mathbf a$ and $\mathbf b$.

Let $\mathbf r$ be the position vector of a point $R$ on $AB$ which divides $AB$ in the ratio $m : n$.



$\mathbf r = \dfrac {n \mathbf a + m \mathbf b} {m + n}$


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.