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.

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:^[1]^[2]^{[better source needed]} ${\begin{array}{rcccccc}A'=&f(a,b,c)&:&g(b,c,a)&:&g(c,a,b)\\B'=&g(a,b,c)&:&f(b,c,a)&:&g(c,a,b)\\C'=&g(a,b,c)&:&g(b,c,a)&:&f(c,a,b)\end{array}}$

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:^[1]^{[better source needed]} ${\begin{array}{rcccccc}A'=&f(a,b,c)&:&g(b,c,a)&:&g(c,b,a)\\B'=&g(a,c,b)&:&f(b,c,a)&:&g(c,a,b)\\C'=&g(a,b,c)&:&g(b,a,c)&:&f(c,a,b)\end{array}}$

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:^[1]^{[better source needed]} ${\begin{array}{rrcrcr}A'=&0\quad \ \ &:&g(b,c,a)&:&-g(c,b,a)\\B'=&-g(a,c,b)&:&0\quad \ \ &:&g(c,a,b)\\C'=&g(a,b,c)&:&-g(b,a,c)&:&0\quad \ \ \end{array}}$

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

Special cases

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.^[3]^{[better source needed]}

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.^[4]^{[better source needed]}

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 $S_{A}={\tfrac {1}{2}}(b^{2}+c^{2}-a^{2}).$ ^[5]^{[better source needed]}

Type 2

Let $X$ be a triangle center. The pedal and antipedal triangles of $X$ are central triangles of Type 2.^[6]^{[better source needed]}
Yff Central Triangle^[7]^{[better source needed]}