Internal and external division points

We find points that divide a segment in a given ratio. Consider the segment joining $A(1, 1)$ and $B(7, 4)$.

An internal division point divides the segment inside, in the given ratio. The point $P$ with $AP : PB = 2 : 1$ is found by mixing the coordinates of $A$ and $B$ with weights $1 : 2$ (the nearer endpoint gets the larger weight).

In general, the point dividing $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m : n$ internally is $\left(\dfrac{n x_1 + m x_2}{m + n}, \dfrac{n y_1 + m y_2}{m + n}\right)$.

An external division point divides on the extension outside the segment. Dividing in the ratio $m : n$ externally is obtained by replacing $n$ with $-n$: $\left(\dfrac{-n x_1 + m x_2}{m - n}, \dfrac{-n y_1 + m y_2}{m - n}\right)$. The point dividing $2 : 1$ externally is $\left(\dfrac{-1 + 14}{1}, \dfrac{-1 + 8}{1}\right) = (13, 7)$, beyond $B$ outside the segment.

The line on the graph passes through $A$ and $B$, and the large dots are $A(1, 1)$, $B(7, 4)$, and the $2 : 1$ internal division point $(5, 3)$.