## What is a Center Function?

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

Kimblerling, C. (2020). Encyclopedia of Triangle Centers. Retrieved September 4, 2020 from: https://faculty.evansville.edu/ck6/encyclopedia/ETC.html

Reinhart, C. et al. Triangle Centers. Retrieved September4, 2020 from: https://brightspace.uakron.edu/d2l/common/viewFile.d2lfile/Database/MTQzMzU4MQ/Presentation_Group6_Reinhart_Kuzas_Burke.pptx?ou=6605&contextId=14871,13579