Ratios of Directed Line Segments Calculator. Enter two endpoints A(x₁, y₁) and B(x₂, y₂) along with a ratio m:n to find the coordinates of the point that partitions the directed line segment. Choose between internal or external division — the calculator returns the exact partition point P(px, py) using the section formula. Also try the find Translated Coordinates with Translation Calculator.
A directed line segment AB is a line segment that has both a defined length and a specific direction — from point A to point B. Unlike a plain line segment, the order of endpoints matters: AB directed from A to B is different from BA directed from B to A. See also our Gradient (Slope) — Gradient.
For internal division, the partition point P is given by P = ((m·x₂ + n·x₁)/(m+n), (m·y₂ + n·y₁)/(m+n)). This is the standard section formula and places P between the two endpoints A and B.
For external division in ratio m:n, the partition point P is P = ((m·x₂ − n·x₁)/(m−n), (m·y₂ − n·y₁)/(m−n)). The point lies outside the segment AB, and m must not equal n to avoid division by zero.
To find the midpoint, use the ratio m:n = 1:1 with internal division. The formula simplifies to P = ((x₁+x₂)/2, (y₁+y₂)/2), which is the standard midpoint formula.
To find the first trisection point (one-third from A), use ratio 1:2 with internal division. For the second trisection point (two-thirds from A), use ratio 2:1. Both points together divide segment AB into three equal parts.
Set the ratio m:n = 1:2 and select internal division. The calculator will return the point that lies exactly one-third of the total segment length away from point A toward point B.
When m = n in external division, the denominator (m − n) becomes zero, making the formula undefined. This means no finite point can divide the segment externally in a 1:1 ratio — the 'point' would be at infinity.
Yes, m and n can be any positive real numbers, not just integers. The section formula works for any ratio — for example, m:n = 1.5:2.5 is perfectly valid and will yield a valid partition point. You might also find our Distance Calculator (3D) useful.