Central triangle

Summary

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 edit

Triangle center function edit

A triangle center function is a real valued function   of three real variables u, v, w having the following properties:

  • Homogeneity property:   for some constant n and for all t > 0. The constant n is the degree of homogeneity of the function  
  • Bisymmetry property:  

Central triangles of Type 1 edit

Let   and   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]

 

Central triangles of Type 2 edit

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 (f, g)-central triangle of Type 2 is a triangle A'B'C' the trilinear coordinates of whose vertices have the following form:[1]

 

Central triangles of Type 3 edit

Let   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]

 

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

Special cases edit

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 edit

Type 1 edit

  • The excentral triangle of triangle ABC is a central triangle of Type 1. This is obtained by taking  
  • Let X be a triangle center defined by the triangle center function   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   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
     
    where S is twice the area of triangle ABC and   [5]

Type 2 edit

References edit

  1. ^ a b c Weisstein, Eric W. "Central Triangle". MathWorld--A Wolfram Web Resource. MathWorld. Retrieved 17 December 2021.
  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--A Wolfram Web Resource. MathWorld. Retrieved 18 December 2021.
  4. ^ Weisstein, Eric W. "Anticevian Triangle". MathWorld--A Wolfram Web Resource. MathWorld. Retrieved 18 December 2021.
  5. ^ Weisstein, Eric W. "Lucas Central Triangle". MathWorld--A Wolfram Web Resource. MathWorld. Retrieved 18 December 2021.
  6. ^ Weisstein, Eric W. "Pedal Triangle". MathWorld--A Wolfram Web Resource. MathWorld. Retrieved 18 December 2021.
  7. ^ Weisstein, Eric W. "Yff Central Triangle". MathWorld--A Wolfram Web Resource. MathWorld. Retrieved 18 December 2021.