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 ⁠$F(u,v,w)$⁠ of three real variables u, v, w having the following properties:

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

Central triangles of Type 1

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

Central triangles of Type 2

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

Central triangles of Type 3

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

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 $f(u,v,w)=-1,\ g(u,v,w)=1.$

• Let X be a triangle center defined by the triangle center function ⁠$g(a,b,c).$⁠ 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 ⁠$f(a,b,c).$⁠ 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 $$f(a,b,c)=a(2S+S_{2}),\quad g(a,b,c)=aS_{A},$$where S is twice the area of triangle ABC and $S_{A}={\tfrac {1}{2}}(b^{2}+c^{2}-a^{2}).$

Type 2

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