Coordinates describing the position of a point in a triangle or simplex. The position of any point P inside, or on the boundary of, a given triangle of reference A₁A₂A₃ can be specified by finding masses m₁, m₂, m₃ such that P is the centre of mass of particles of masses m₁, m₂, m₃ placed at A₁, A₂, A₃ respectively. In this case P is said to have barycentric coordinates (m₁, m₂, m₃). For an internal point of the triangle the masses are all positive. If k>0 then (km₁, km₂, km₃) represents the same point, so barycentric coordinates are usually chosen so that m₁+m₂+m₃=1. With this convention, the barycentric coordinates of A₁ are (1, 0, 0), and those of the midpoint of A₁A₂ are (, , 0). Suppose a random trial can result in one of three possibilities with probabilities p₁, p₂, p₃, such that p₁+p₂+p₃=1, then allocation of these probabilities can be represented by a point inside, or on the boundary of, the triangle. The triangular display is sometimes referred to as a ternary diagram. The idea of barycentric coordinates can be generalized to the case of a tetrahedron in three dimensions or a simplex. See also areal coordinates.

Barycentric coordinates. Illustrated for compositions of three classes having proportions x, y, and z. The case where the three classes are equally likely corresponds to the centroid of the equilateral triangle.