Triangle Center Function

A triangle center function (sometimes simply called a center function) is a nonzero function f(a,b,c) that is homogeneous


bisymmetry in b and c,


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


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




respectively (Kimberling 1998, p. 46).

Because of the symmetry in the definition of trilinear coordinates, a single function alpha suffices to determine all three coordinates of a center simply through cyclical permutation of the variables. These variables may correspond to angles (A, B, C), side lengths (a, b, c), 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 G can be given by


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


Two triangle center functions for a single triangle center need not be identical. For example, if h_a is the a-altitude of triangle DeltaABC, then the expressions cscA, sinBsinC, 1/a, bc, and h_a are equivalent triangle center functions for the triangle centroid G, even though cscA!=sinBsinC. Two triangle center functions are equivalent (i.e., they are triangle functions of the same center) iff their ratio is a function symmetric in a, b and c and/or A, B, and C. For example, the ratio of the centroid's triangle functions cscA and h_a is cscA/h_a=1/(2RsinAsinBsinC), where R is the circumradius of DeltaABC. Hence, they are equivalent triangle center functions.

Note also that it is common to give triangle center functions in an abbreviated form f^'(a,b,c) that does not explicitly satisfy bisymmetry, but rather biantisymmetry, so f^'(a,c,b)=-f^'(a,b,c). In such cases, f^'(a,b,c) can be converted to an equivalent form f(a,b,c) that does satisfy the bisymmetry property by defining


An example of this kind is Kimberling center X_(100), which has a tabulated center of


which corresponds to the true triangle center function


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

See also

Exact Trilinear Coordinates, Kimberling Center, Major Triangle Center, Regular Triangle Center, Triangle Center, Trilinear Coordinates

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Bottema, O. "Het begrip 'merkwaardig' met betrekking tot punten in de driehoeksmeetkunde." Nieuw Tijdschr. Wisk., 69, 2-7, 1981-82.Kimberling, C. "Triangle Centers.", C. "Triangle Centers as Functions." Rocky Mtn. J. Math. 23, 1269-1286, 1993.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Lester, J. "Triangles III: Complex Triangle Functions." Aequationes Math. 53, 4-35, 1997.

Referenced on Wolfram|Alpha

Triangle Center Function

Cite this as:

Weisstein, Eric W. "Triangle Center Function." From MathWorld--A Wolfram Web Resource.

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