Triangle Center

A triangle center (sometimes simply called a center) is a point whose trilinear coordinates are defined in terms of the side lengths and angles of a triangle and for which a triangle center function can be defined. The function giving the coordinates is called the triangle center function. The four ancient centers are the triangle centroid, incenter, circumcenter, and orthocenter.

The triangle center functions of triangles centers therefore satisfy homogeneity

(1)

bisymmetry in and ,

(2)

and cyclicity in , , and ,

(3)

(Kimberling 1998, p. 46).

Note that most, but not all, special triangle points therefore qualify as triangle centers. For example, bicentric points fail to satisfy bisymmetry, and are therefore excluded. The most common examples of points of this type are the first and second Brocard points, for which triangle center-like functions can be defined that obey homogeneity and cyclicity, but not bisymmetry.

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

(4)

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

(5)

which corresponds to the true triangle center function

(6)

A triangle center is said to be polynomial iff there is a triangle center function that is a polynomial in , , and (Kimberling 1998, p. 46).

Similarly, a triangle center is said to be regular iff there is a triangle center function that is a polynomial in , , , and , where is the area of the triangle).

A triangle center is said to be a major triangle center if the triangle center function is a function of angle alone, and therefore and of and alone, respectively.

C. Kimberling (1998) has extensively tabulated triangle centers and their trilinear coordinates, assigning a unique integer to each. In this work, these centers are called Kimberling centers, and the th center is denoted , the first few of which are summarized below.

	center	triangle center function
	incenter	1
	triangle centroid	, ,
	circumcenter	,
	orthocenter
	nine-point center	, ,
	symmedian point	,
	Gergonne point	,
	Nagel point	,
	mittenpunkt	,
	Spieker center
	Feuerbach point	,
	harmonic conjugate of with respect to and	, ,
	first Fermat point	,
	second Fermat point	,
	first isodynamic point	,
	second isodynamic point	,
	first Napoleon point	,
	second Napoleon point	,
	Clawson point	, , ,
	de Longchamps point

E. Brisse has compiled a separate list of 2001 triangle centers.