Central triangle explained

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 of three real variables having the following properties:

Homogeneity property

F(tu,tv,tw)=tⁿF(u,v,w)

for some constant and for all . The constant is the degree of homogeneity of the function

Bisymmetry property:

F(u,v,w)=F(u,w,v).

Central triangles of Type 1

Let and be two triangle center functions, not both identically zero functions, having the same degree of homogeneity. Let be the side lengths of the reference triangle . An -central triangle of Type 1 is a triangle the trilinear coordinates of whose vertices have the following form:^[1] ^[2] $\begin A' =& f(a,b,c) &:& g(b,c,a) &:& g(c,a,b) \\ B' =& g(a,b,c) &:& f(b,c,a) &:& g(c,a,b) \\ C' =& g(a,b,c) &:& g(b,c,a) &:& f(c,a,b)\end$

Central triangles of Type 2

Let be a triangle center function and be a function function satisfying the homogeneity property and having the same degree of homogeneity as but not satisfying the bisymmetry property. An -central triangle of Type 2 is a triangle the trilinear coordinates of whose vertices have the following form:^[1] $\begin A' =& f(a,b,c) &:& g(b,c,a) &:& g(c,b,a) \\ B' =& g(a,c,b) &:& f(b,c,a) &:& g(c,a,b) \\ C' =& g(a,b,c) &:& g(b,a,c) &:& f(c,a,b)\end$

Central triangles of Type 3

Let be a triangle center function. An -central triangle of Type 3 is a triangle the trilinear coordinates of whose vertices have the following form:^[1] $\begin A' =& 0 \quad\ \ &:& g(b,c,a) &:& - g(c,b,a) \\ B' =& - g(a,c,b) &:& 0 \quad\ \ &:& g(c,a,b) \\ C' =& g(a,b,c) &:& - g(b,a,c) &:& 0 \quad\ \ \end$

This is a degenerate triangle in the sense that the points are collinear.

Special cases

If, the -central triangle of Type 1 degenerates to the triangle center . 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 is a central triangle of Type 1. This is obtained by taking

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

Let be a triangle center defined by the triangle center function Then the cevian triangle of is a -central triangle of Type 1.^[3]
Let be a triangle center defined by the triangle center function Then the anticevian triangle of is a -central triangle of Type 1.^[4]
The Lucas central triangle is the -central triangle with

f(a,b,c) = a(2S+S_2), \quad g(a,b,c) = aS_A, where is twice the area of triangle ABC and

S_A=\tfrac{1}{2}(b²+c²-a^2).

^[5]

Type 2

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

Notes and References

Web site: Weisstein, Eric W . Central Triangle . MathWorld--A Wolfram Web Resource. . MathWorld . 17 December 2021.
Kimberling, C . Triangle Centers and Central Triangles . Congressus Numerantium. A Conference Journal on Numerical Themes. 129 . 1998 . 129.
Web site: Weisstein, Eric W . Cevian Triangle . MathWorld--A Wolfram Web Resource. . MathWorld . 18 December 2021.
Web site: Weisstein, Eric W . Anticevian Triangle . MathWorld--A Wolfram Web Resource. . MathWorld . 18 December 2021.
Web site: Weisstein, Eric W . Lucas Central Triangle . MathWorld--A Wolfram Web Resource. . MathWorld . 18 December 2021.
Web site: Weisstein, Eric W . Pedal Triangle . MathWorld--A Wolfram Web Resource. . MathWorld . 18 December 2021.
Web site: Weisstein, Eric W . Yff Central Triangle . MathWorld--A Wolfram Web Resource. . MathWorld . 18 December 2021.