Section Formula (Internal Division)

Often what is needed is not the distance between two points but a point that lies between them in a stated proportion — for instance, a warehouse located one-third of the way from a firm’s old depot to its new one. If a point $P$ lies on the segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ and divides it internally in the ratio $m : n$ (meaning $AP : PB = m : n$), the coordinates of $P$ are given by the section formula.

Notice which endpoint’s coordinates get which weight: $B$’s coordinates are weighted by $m$ (the part of the ratio measured from $A$), and $A$’s coordinates are weighted by $n$ (the part of the ratio measured from $B$). It is easy to swap these weights by mistake, so always re-check which end of the segment the dividing point sits closer to before substituting numbers into the formula.

A very common special case is the mid-point of a segment, where $P$ divides $AB$ in the ratio $1:1$ (equally). Putting $m = n = 1$ collapses the section formula to a simple average of the two endpoints: …