Central triangle

Short description: Triangle related to a given triangle by two functions

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 [math]\displaystyle{ F(u,v,w) }[/math] of three real variables u, v, w having the following properties:

• Homogeneity property: [math]\displaystyle{ F(tu,tv,tw) = t^n F(u,v,w) }[/math] for some constant n and for all t > 0. The constant n is the degree of homogeneity of the function [math]\displaystyle{ F(u,v,w). }[/math]
• Bisymmetry property: [math]\displaystyle{ F(u,v,w) = F(u,w,v). }[/math]

Central triangles of Type 1

Let [math]\displaystyle{ f(u,v,w) }[/math] and [math]\displaystyle{ g(u,v,w) }[/math] 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: [math]\displaystyle{ \begin{array}{rcccccc} A' =& f(a,b,c) &:& g(b,c,a) &:& g(c,a,b) \\ B' =& g(a,b,c) &:& f(b,c,a) &:& g(c,a,b) \\ C' =& g(a,b,c) &:& g(b,c,a) &:& f(c,a,b) \end{array} }[/math]

Central triangles of Type 2

Let [math]\displaystyle{ f(u,v,w) }[/math] be a triangle center function and [math]\displaystyle{ g(u,v,w) }[/math] be a function function satisfying the homogeneity property and having the same degree of homogeneity as [math]\displaystyle{ f(u,v,w) }[/math] 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: [math]\displaystyle{ \begin{array}{rcccccc} A' =& f(a,b,c) &:& g(b,c,a) &:& g(c,b,a) \\ B' =& g(a,c,b) &:& f(b,c,a) &:& g(c,a,b) \\ C' =& g(a,b,c) &:& g(b,a,c) &:& f(c,a,b) \end{array} }[/math]

Central triangles of Type 3

Let [math]\displaystyle{ g(u,v,w) }[/math] 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: [math]\displaystyle{ \begin{array}{rrcrcr} A' =& 0 \quad\ \ &:& g(b,c,a) &:& - g(c,b,a) \\ B' =& - g(a,c,b) &:& 0 \quad\ \ &:& g(c,a,b) \\ C' =& g(a,b,c) &:& - g(b,a,c) &:& 0 \quad\ \ \end{array} }[/math]

This is a degenerate triangle in the sense that the points A', B', C' are collinear.

Special cases

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.

Examples

Type 1

• The excentral triangle of triangle △ABC is a central triangle of Type 1. This is obtained by taking [math]\displaystyle{ f(u,v,w) = -1,\ g(u,v,w) = 1. }[/math]

• Let X be a triangle center defined by the triangle center function [math]\displaystyle{ g(a,b,c). }[/math] Then the cevian triangle of X is a (0, g)-central triangle of Type 1.

• Let X be a triangle center defined by the triangle center function [math]\displaystyle{ f(a,b,c). }[/math] Then the anticevian triangle of X is a (−f, f)-central triangle of Type 1.

• The Lucas central triangle is the (f, g)-central triangle with [math]\displaystyle{ f(a,b,c) = a(2S+S_2), \quad g(a,b,c) = aS_A, }[/math]where S is twice the area of triangle ABC and [math]\displaystyle{ S_A = \tfrac{1}{2}(b^2 + c^2 - a^2). }[/math]

Type 2

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