Central Triangle

Let each of and be a triangle center function or the zero function, and let one of the following three conditions hold.

1. The degree of homogeneity of equals that of .

2. is the zero function and is not the zero function.

3. is the zero function and is not the zero function.

Also define three points with the following trilinear coordinates.

			(1)
			(2)
			(3)

Then is said to be an -central triangle of type 1, and any triangle for which these equations hold for some choice of triangle center functions is called a central triangle of type 1. Such a triangle is completely determined by its first vertex , but has complete trilinear vertex matrix given by

(4)

If , then and the triangle degenerates to the triangle center ; otherwise, is nondegenerate (Kimberling 1998, p. 54). Cevian and anticevian triangles are both of type 1.

If fails to be bicentric so that , then the resulting triangle determined by and is known as an -central triangle of type 2 and has trilinear vertex matrix

(5)

No triangle of type 2 is also of type 2. Pedal and antipedal triangles are both of type 2.

Given only a single center function , then the -central triangle of type 3 is the degenerate triangle with collinear vertices given by trilinear vertex matrix

(6)

This "triangle" is also called the cocevian triangle of the center (Kimberling 1998, p. 54).

All triangle centers of an equilateral central triangle degenerate into a single center of the reference triangle.