Internal and external division points

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

Internal division

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 getting the larger weight.

P = \left(\frac{1 \cdot 1 + 2 \cdot 7}{2 + 1}, \ \frac{1 \cdot 1 + 2 \cdot 4}{2 + 1}\right) = (5, 3)

In general, the point dividing $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m : n$ internally is the following.

\left(\frac{n x_1 + m x_2}{m + n}, \frac{n y_1 + m y_2}{m + n}\right)

Why the weights look backwards

Thinking of the journey from $A$ to $B$ makes them natural. Saying $AP : PB = 2 : 1$ means $P$ sits $\dfrac{2}{3}$ of the way along.

\begin{align*} x &= 1 + \frac{2}{3}(7 - 1) = 5 \\ y &= 1 + \frac{2}{3}(4 - 1) = 3 \end{align*}

The closer the point is to $B$ , the more strongly the coordinates of $B$ count, which is why $B$ carries the larger weight.

Three cases

Ratio	Point	Position
$1 : 1$ internal	$\left(4, \dfrac{5}{2}\right)$	the midpoint
$2 : 1$ internal	$(5, 3)$	inside the segment
$2 : 1$ external	$(13, 7)$	beyond $B$

External division

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(\frac{-n x_1 + m x_2}{m - n}, \frac{-n y_1 + m y_2}{m - n}\right)

Only when $m = n$ does the denominator vanish and no external point exist: dividing $1 : 1$ externally would place the point infinitely far along the line.

Applications

The centroid of a triangle divides each median in the ratio $2 : 1$
Linear interpolation estimates a value between two known points¹
Bezier curves are built from repeated internal division

The line on the graph passes through $A$ and $B$ , and the large dots are $A$ , $B$ , and the $2 : 1$ internal division point.

Linear interpolation, Wikipedia