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

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.

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 $(&minus;f, f)$-central triangle of Type 1.

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).$$
 * The Lucas central triangle is the $(f, g)$-central triangle with $$

Type 2

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