Triangle Center Function

A triangle center function (sometimes simply called a center function) is a nonzero function that is homogeneous

(1)

bisymmetry in and ,

(2)

and such that the trilinear coordinates of the triangle center encoded by the function are cyclic in , , and ,

(3)

These three properties are satisfied by almost all common special points of triangles (Bottema, 1981-82), and may be referred to as homogeneity, bisymmetry, and cyclicity, respectively (Kimberling 1998, p. 46).

This definition is grounded in the geometric properties shared by virtually all special points of planar triangles. However, an important exception are bicentric points, which lack the bisymmetry property, and are therefore not triangle centers. The best known example of points of this type are the first and second Brocard points, which have trilinear coordinates

(4)

and

(5)

respectively (Kimberling 1998, p. 46).

Because of the symmetry in the definition of trilinear coordinates, a single function suffices to determine all three coordinates of a center simply through cyclical permutation of the variables. These variables may correspond to angles (, , ), side lengths (, , ), or a mixture, since side lengths and angles can be interconverted using the law of cosines.

For example, the triangle center function for a triangle centroid can be given by

(6)

where the sides of the triangle have lengths , , and . Cyclically permuting the variables then gives the full trilinear coordinates of the centroid as

(7)

Two triangle center functions for a single triangle center need not be identical. For example, if is the -altitude of triangle , then the expressions , , , , and are equivalent triangle center functions for the triangle centroid , even though . Two triangle center functions are equivalent (i.e., they are triangle functions of the same center) iff their ratio is a function symmetric in , and and/or , , and . For example, the ratio of the centroid's triangle functions and is , where is the circumradius of . Hence, they are equivalent triangle center functions.

Note also that it is common to give triangle center functions in an abbreviated form that does not explicitly satisfy bisymmetry, but rather biantisymmetry, so . In such cases, can be converted to an equivalent form that does satisfy the bisymmetry property by defining

(8)

An example of this kind is Kimberling center , which has a tabulated center of

(9)

which corresponds to the true triangle center function

(10)

Kimberling (1994, 1998, and online) has enumerated thousands of triangle centers, known in this work as Kimberling centers in his honor, with the th Kimberling center being denoted .