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.
Definitionedit
Triangle center functionedit
A triangle center function is a real valued function of three real variables u, v, w having the following properties:
Homogeneity property: for some constant n and for all t > 0. The constant n is the degree of homogeneity of the function
Bisymmetry property:
Central triangles of Type 1edit
Let and be two triangle center functions, not both identically zero functions, having the same degree of homogeneity. Let a, b, c be the side lengths of the reference triangle △ABC. An (f, g)-central triangle of Type 1 is a triangle △A'B'C' the trilinear coordinates of whose vertices have the following form:[1][2]
Central triangles of Type 2edit
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 (f, g)-central triangle of Type 2 is a triangle △A'B'C' the trilinear coordinates of whose vertices have the following form:[1]
Central triangles of Type 3edit
Let be a triangle center function. An g-central triangle of Type 3 is a triangle △A'B'C' the trilinear coordinates of whose vertices have the following form:[1]
This is a degenerate triangle in the sense that the points A', B', C' are collinear.
Special casesedit
If f = g, the (f, g)-central triangle of Type 1 degenerates to the triangle center A'. All central triangles of both Type 1 and Type 2 relative to an equilateral triangle degenerate to a point.
Examplesedit
Type 1edit
The excentral triangle of triangle △ABC is a central triangle of Type 1. This is obtained by taking
Let X be a triangle center defined by the triangle center function Then the cevian triangle of X is a (0, g)-central triangle of Type 1.[3]
Let X be a triangle center defined by the triangle center function Then the anticevian triangle of X is a (−f, f)-central triangle of Type 1.[4]
The Lucas central triangle is the (f, g)-central triangle with