Central Triangle

A central triangle is a triangle specified relative to a reference triangle by one of the following three trilinear vertex matrix constructions.

For type 1, 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.

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

If , then and the triangle degenerates to the triangle center (Kimberling 1998, p. 54). Other choices can also give degenerate configurations when the trilinear vertex matrix is a singular matrix. Cevian and anticevian triangles are both of type 1.

If the symmetry condition on is relaxed so that , then the resulting triangle determined by and is known as an -central triangle of type 2 and has trilinear vertex matrix

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

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.

The following table compares central triangles. Here , , and are the side lengths of the reference triangle, is its semiperimeter, and , , and also denote its angles. The last column gives the trilinear coordinates of the first polygon vertex ; the other two are obtained by cyclically permuting the vertices, side lengths, angles, and coordinate positions. The drawings use the acute triangle with Cartesian coordinates , , and , shown in gray, with the derived triangle in blue. Each drawing is independently scaled. The orthic triangle and tangential triangle rows assume that no angle of the reference triangle is a right angle.

For the Apollonius triangle row, write , , and , where

Both are homogeneous of degree 6 and symmetric in their last two arguments, so they are triangle center functions of the same degree. Multiplying the three rows of the trilinear vertex matrix of the Apollonius triangle by , , and , respectively, gives the type 1 form above (Lozada 2025).