Midpoints and section ratios

The midpoint of (x₁,y₁) and (x₂,y₂) averages each coordinate separately. For a point P dividing AB internally in ratio AP:PB = m:n, its position is A + m/(m+n)(B − A). The fraction uses the distance travelled from A, which helps prevent the weights being reversed.

An internal point lies between the endpoints. Check each coordinate against the endpoint coordinates, allowing equality when that coordinate is unchanged along the segment. A coordinate outside those bounds suggests either an external division or an error. The ratio describes lengths along the same straight line; it is not a ratio of the x-coordinates alone.

Worked example

A = (−2,1), B = (8,6), and AP:PB = 2:3. Find P.

1. P lies 2/(2+3) = 2/5 of the way from A to B.
2. B − A = (10,5), so the displacement from A is (4,2).
3. Add this displacement to A.

Answer: P = (2,3)