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 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:

Central triangles of Type 3 
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.

Special cases
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.

Examples

Type 1 
The excentral triangle of triangle  is a central triangle of Type 1. This is obtained by taking 

Let  be a triangle center defined by the triangle center function  Then the cevian triangle of  is a -central triangle of Type 1.

Let  be a triangle center defined by the triangle center function  Then the anticevian triangle of  is a -central triangle of Type 1.

The Lucas central triangle is the -central triangle with where  is twice the area of triangle ABC and

Type 2 
Let  be a triangle center. The pedal and antipedal triangles of  are central triangles of Type 2.
Yff Central Triangle

References

Triangle geometry