**Trilinear coordinates**, introduced by Möbius in 1827, is a way to identify a point in a plane with respect to a triangle. The coordinates are defined by measuring signed distances from a triangle’s sides.

Basically, it is a coordinate system where a triangle assigns three coordinates to a point P [1]:

- α = [P, BC]
- β = [P, AC]
- γ = [P, AB].

The three numbers are an ordered triple (α, β, γ), where each is proportional to the directed distance from P to one of the triangle’s side lines [2].

## Example of Use

Trilinear coordinates are used to study the geometric properties of triangles using algebra. In real life, let’s say you are studying a mixture of three components a, b, and c. Point P could represent a mixture with percent composition represented by the trilinear coordinates (α, β, γ).

Let’s say a triangle ABC has a perpendicular from vertex A to base BC, as shown in the above image. If the perpendicular A = 100 units (representing 100 percent of the mixture), then a point P with trilinear coordinates (α, β, γ) will satisfy α + β + γ = 100.

## Trilinear Coordinates vs. Barycentric Coordinates

Other names for trilinear coordinates include *trilinears*, *homogeneous trilinear coordinates*, and *barycentric coordinates* (a *barycenter *is another name for center of mass). Although the names are often used synonymously, there is a subtle difference between barycentric coordinates and trilinears [3]:

- The barycentric coordinates of point P are an ordered triple where P is the centroid consisting of three masses at vertices A, B, and C.
- Trilinear coordinates of point P are an ordered triple that is proportional to the directed distance from P to the sides.

## The Center Function and Trilinear Coordinates

A center function (also called a *triangle center function*, *symmetric triangle center function* or simply a *center*) gives the trilinear coordinates of a triangle’s center. The function’s three variables {a, b, c} or {α, β γ} correspond to angles or sides.

## What is a Triangle Center?

A **triangle center** is a point defined in terms of a triangle’s side lengths and angles and for which a center function exists. A **major triangle center** is one where the function α = f(A, B, C) is only a function of angle A.

It sounds simple, but don’t be fooled into thinking there’s just one center to a triangle. A “center” can be defined in many ways. For example, the *centroid* is the intersection of the triangle medians, and the *circumcenter* is the center of the circle inscribing the triangle.

In fact, there are thousands of different triangle centers including the Fermat point, which has the center function α = csc(A + ⅓π) and the Far-Out point, with function α = a(b^{4} + c^{4} – a^{4} – b^{2}c^{2}).

## Properties of the Triangle Center Function

The triangle center function is a homogeneous function (i.e. with variables that increase by the same proportion). In notation, that’s:

**f ( t a, t b, t c) = t ^{n} f (a, b, c).**

Where

*n*is a constant.

The function is also **nonzero**, meaning that it can’t have values equal to zero. This makes sense, because a non-zero value of a triangle would result in a non-triangle shape: a line, a point, or no shape at all.

The triangle center function has bisymmetry (two planes of symmetry at right angles) in the second and third variables. In notation:

f(a, b, c) = f(a, c, b).

## References

[1] 22 Trilinear Coordinates. Retrieved May 15, 2022 from: http://www.mcs.uvawise.edu/msh3e/resources/geometryBook/22TrilinearCoordinates.pdf

[2] Yiu, P. The uses of homogeneous barycentric coordinates in plane Euclidean geometry.

[3] Boju, V. & Funar, L. (2007). The Math Problems Notebook. Birkhäuser Boston