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:

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