Section formula

In coordinate geometry, Section formula is 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 a point P (lying on AB) divides AB in the ratio m:n then
formula_1
The ratio m:n can also be written as formula_2, or formula_3, where formula_4. So, the coordinates of point P dividing the line segment joining the points A"(x1,y1)" and B"(x2,y2)" are:
formula_5
formula_6
formula_7
Similarly, the ratio can also be written as formula_8, and the coordinates of P are formula_9.
External Divisions.
If a point P (lying on the extension of AB) divides AB in the ratio m:n then
formula_10
Midpoint formula.
The midpoint of a line segment divides it internally in the ratio formula_11. Applying the Section formula for internal division:
formula_12
Derivation.
formula_13
formula_14
formula_15
In 3-Dimensions.
Let A and B be two points with Cartesian coordinates "(x1, y1, z1)" and "(x2, y2, z2)" and P be a point on the line through A and B. If formula_16. Then the section formulae give the coordinates of P as
formula_17
If, instead, P is a point on the line such that formula_18, its coordinates are formula_19.
In vectors.
The position vector of a point P dividing the line segment joining the points A and B whose position vectors are formula_20 and formula_21 
External links.
section-formula by GeoGebra