Triangle center function

A triangle center function is a real valued function ${\displaystyle F(u,v,w)}$ of three real variables u, v, w having the following properties:

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

Central triangles of Type 1

Let ${\displaystyle f(u,v,w)}$ and ${\displaystyle g(u,v,w)}$ 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]

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

Central triangles of Type 2

Let ${\displaystyle f(u,v,w)}$ be a triangle center function and ${\displaystyle g(u,v,w)}$ be a function function satisfying the homogeneity property and having the same degree of homogeneity as ${\displaystyle f(u,v,w)}$ 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]

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

Central triangles of Type 3

Let ${\displaystyle g(u,v,w)}$ 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]

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

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

Type 1

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

• Let X be a triangle center defined by the triangle center function ${\displaystyle g(a,b,c).}$ Then the cevian triangle of X is a (0, g)-central triangle of Type 1.^[3]

• Let X be a triangle center defined by the triangle center function ${\displaystyle f(a,b,c).}$ Then the anticevian triangle of X is a (−f, f)-central triangle of Type 1.^[4]

• The Lucas central triangle is the (f, g)-central triangle with
  $${\displaystyle f(a,b,c)=a(2S+S_{2}),\quad g(a,b,c)=aS_{A},}$$

  
  where S is twice the area of triangle ABC and ${\displaystyle S_{A}={\tfrac {1}{2}}(b^{2}+c^{2}-a^{2}).}$ ^[5]

Type 2

• Let X be a triangle center. The pedal and antipedal triangles of X are central triangles of Type 2.^[6]
• Yff Central Triangle^[7]