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.

Definition

Triangle center function

A triangle center function is a real valued function Template:Tmath of three real variables Template:Mvar having the following properties:

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

Central triangles of Type 1

Let Template:Tmath and Template:Tmath be two triangle center functions, not both identically zero functions, having the same degree of homogeneity. Let Template:Mvar be the side lengths of the reference triangle Template:Math. An Template:Math-central triangle of Type 1 is a triangle Template:Math the trilinear coordinates of whose vertices have the following form:Template:Bcn $$\begin{array}{cccccc}{A}^{\prime }=& f(a,b,c)& :& g(b,c,a)& :& g(c,a,b)\\ B^{\prime }=& g(a,b,c)& :& f(b,c,a)& :& g(c,a,b)\\ C^{\prime }=& g(a,b,c)& :& g(b,c,a)& :& f(c,a,b)\end{array}$$

Central triangles of Type 2

Let Template:Tmath be a triangle center function and Template:Tmath be a function function satisfying the homogeneity property and having the same degree of homogeneity as Template:Tmath but not satisfying the bisymmetry property. An Template:Math-central triangle of Type 2 is a triangle Template:Math the trilinear coordinates of whose vertices have the following form:Template:Bcn $$\begin{array}{cccccc}{A}^{\prime }=& f(a,b,c)& :& g(b,c,a)& :& g(c,b,a)\\ B^{\prime }=& g(a,c,b)& :& f(b,c,a)& :& g(c,a,b)\\ C^{\prime }=& g(a,b,c)& :& g(b,a,c)& :& f(c,a,b)\end{array}$$

Central triangles of Type 3

Let Template:Tmath be a triangle center function. An Template:Mvar-central triangle of Type 3 is a triangle Template:Math the trilinear coordinates of whose vertices have the following form:Template:Bcn $$\begin{array}{cccccc}{A}^{\prime }=& 0& :& g(b,c,a)& :& -g(c,b,a)\\ B^{\prime }=& -g(a,c,b)& :& 0& :& g(c,a,b)\\ C^{\prime }=& g(a,b,c)& :& -g(b,a,c)& :& 0\end{array}$$

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

Special cases

If Template:Math, the Template:Math-central triangle of Type 1 degenerates to the triangle center Template:Mvar. 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 Template:Math is a central triangle of Type 1. This is obtained by taking $f(u,v,w)=-1,g(u,v,w)=1.$

• Let Template:Mvar be a triangle center defined by the triangle center function Template:Tmath Then the cevian triangle of Template:Mvar is a Template:Math-central triangle of Type 1.Template:Bcn

• Let Template:Mvar be a triangle center defined by the triangle center function Template:Tmath Then the anticevian triangle of Template:Mvar is a Template:Math-central triangle of Type 1.Template:Bcn

• The Lucas central triangle is the Template:Math-central triangle with $$f(a,b,c)=a(2S+S_2),g(a,b,c)=a{S}_A,$$where Template:Mvar is twice the area of triangle ABC and $S_A=\frac{1}{2}(b^2+c^2-a^2).$ Template:Bcn

Type 2

• Let Template:Mvar be a triangle center. The pedal and antipedal triangles of Template:Mvar are central triangles of Type 2.Template:Bcn
• Yff Central TriangleTemplate:Bcn