Central triangle

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Short description: Triangle related to a given triangle by two functions

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 F(u,v,w) of three real variables u, v, w having the following properties:

  • Homogeneity property: F(tu,tv,tw)=tnF(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:[1][2] 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)

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:[1] 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)

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:[1] A′=0  :g(b,c,a):−g(c,b,a)B′=−g(a,c,b):0  :g(c,a,b)C′=g(a,b,c):−g(b,a,c):0  

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.

Examples

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.[3]
  • Let X be a triangle center defined by the triangle center function 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 f(a,b,c)=a(2S+S2),g(a,b,c)=aSA,where S is twice the area of triangle ABC and SA=12(b2+c2−a2). [5]

Type 2

References

  1. ↑ 1.0 1.1 1.2 Weisstein, Eric W. "Central Triangle". MathWorld. https://mathworld.wolfram.com/CentralTriangle.html. 
  2. ↑ Kimberling, C (1998). "Triangle Centers and Central Triangles". Congressus Numerantium. A Conference Journal on Numerical Themes. 129 129. 
  3. ↑ Weisstein, Eric W. "Cevian Triangle". MathWorld. https://mathworld.wolfram.com/CevianTriangle.html. 
  4. ↑ Weisstein, Eric W. "Anticevian Triangle". MathWorld. https://mathworld.wolfram.com/AnticevianTriangle.html. 
  5. ↑ Weisstein, Eric W. "Lucas Central Triangle". MathWorld. https://mathworld.wolfram.com/LucasCentralTriangle.html. 
  6. ↑ Weisstein, Eric W. "Pedal Triangle". MathWorld. https://mathworld.wolfram.com/PedalTriangle.html. 
  7. ↑ Weisstein, Eric W. "Yff Central Triangle". MathWorld. https://mathworld.wolfram.com/YffCentralTriangle.html. 



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