Central triangle
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. Definition Triangle center function A triangle center function is a real valued function of three real variables having the following properties: Homogeneity property: for some constant and for all . The constant is the degree of homogeneity of the function Bisymmetry property: Central triangles of Type 1 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: Central triangles of Type 2 Let be a triangle center function and be a function function satisfying the homogeneity property and having