Section formula


In coordinate geometry, the Section formula is a formula used to find the ratio in which a line segment is divided by a point internally or externally. It is used to find out the centroid, incenter and excenters of a triangle. In physics, it is used to find the center of mass of systems, equilibrium points, etc.

Internal divisions

If point P divides the line segment AB joining the points and in the ratio m:n, then
The ratio m:n can also be written as, or, where. So, the coordinates of point dividing the line segment joining the points and are:
Similarly, the ratio can also be written as, and the coordinates of P are.

Proof

Triangles.

External divisions

If a point P divides AB in the ratio m:n then

Proof

Triangles .
Therefore ∠ACP = ∠BDP

Midpoint formula

The midpoint of a line segment divides it internally in the ratio. Applying the Section formula for internal division:

Centroid

The centroid of a triangle is the intersection of the medians and divides each median in the ratio. Let the vertices of the triangle be, and. So, a median from point A will intersect BC at.
Using the section formula, the centroid becomes:

In three dimensions

Let A and B be two points with Cartesian coordinates ' and ' and P be a point on the line through A and B. If. Then the section formula gives the coordinates of P as
If, instead, P is a point on the line such that, its coordinates are.

In vectors

The position vector of a point P dividing the line segment joining the points A and B whose position vectors are and
  1. in the ratio internally, is given by
  2. in the ratio externally, is given by