In geometry, a central triangle is a triangle in the plane of the reference triangle. The trilinear coordinates of its vertices relative to the reference triangle are expressible in a certain cyclical way in terms of two functions having the same degree of homogeneity. At least one of the two functions must be a triangle center function. The excentral triangle is an example of a central triangle. The central triangles have been classified into three types based on the properties of the two functions.
A triangle center function is a real valued function of three real variables having the following properties:
Let and be two triangle center functions, not both identically zero functions, having the same degree of homogeneity. Let be the side lengths of the reference triangle . An -central triangle of Type 1 is a triangle the trilinear coordinates of whose vertices have the following form:
Let be a triangle center function and be a function function satisfying the homogeneity property and having the same degree of homogeneity as but not satisfying the bisymmetry property. An -central triangle of Type 2 is a triangle the trilinear coordinates of whose vertices have the following form:
Let be a triangle center function. An -central triangle of Type 3 is a triangle the trilinear coordinates of whose vertices have the following form:
This is a degenerate triangle in the sense that the points are collinear.
If , the -central triangle of Type 1 degenerates to the triangle center . All central triangles of both Type 1 and Type 2 relative to an equilateral triangle degenerate to a point.