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Central triangle

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Triangle related to a given triangle by two functions
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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 ) {\displaystyle F(u,v,w)} ⁠ of three real variables u, v, w having the following properties:

  • Homogeneity property: F ( t u , t v , t w ) = t n F ( u , v , w ) {\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 ⁠ F ( u , v , w ) . {\displaystyle F(u,v,w).} ⁠
  • Bisymmetry property: F ( u , v , w ) = F ( u , w , v ) . {\displaystyle F(u,v,w)=F(u,w,v).}

Central triangles of Type 1

Let ⁠ f ( u , v , w ) {\displaystyle f(u,v,w)} ⁠ and ⁠ g ( u , v , w ) {\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: 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 ) {\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}}}

Central triangles of Type 2

Let ⁠ f ( u , v , w ) {\displaystyle f(u,v,w)} ⁠ be a triangle center function and ⁠ g ( u , v , w ) {\displaystyle g(u,v,w)} ⁠ be a function function satisfying the homogeneity property and having the same degree of homogeneity as ⁠ f ( u , v , w ) {\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: 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 ) {\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}}}

Central triangles of Type 3

Let ⁠ g ( u , v , w ) {\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: 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     {\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.

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. {\displaystyle 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 ) . {\displaystyle g(a,b,c).} ⁠ Then the cevian triangle of X is a (0, g)-central triangle of Type 1.
  • Let X be a triangle center defined by the triangle center function ⁠ f ( a , b , c ) . {\displaystyle f(a,b,c).} ⁠ Then the anticevian triangle of X is a (−f, f)-central triangle of Type 1.
  • The Lucas central triangle is the (f, g)-central triangle with f ( a , b , c ) = a ( 2 S + S 2 ) , g ( a , b , c ) = a S A , {\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 S A = 1 2 ( b 2 + c 2 − a 2 ) . {\displaystyle S_{A}={\tfrac {1}{2}}(b^{2}+c^{2}-a^{2}).}

Type 2

References

  1. ^ 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.
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