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.

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: ${\displaystyle F(u,v,w)=F(u,w,v).}$

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

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

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

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

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.

• 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][better source needed]}

• 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][better source needed]}

• 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][better source needed]}

• 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]}