Central triangle

{{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 {{tmath|F(u,v,w)}} of three real variables {{mvar|u, v, w}} having the following properties:

:*Homogeneity property: $F(tu,tv,tw) = t^n F(u,v,w)$ for some constant {{mvar|n}} and for all {{math|t > 0}}. The constant {{mvar|n}} is the degree of homogeneity of the function {{tmath|F(u,v,w).}}

:*Bisymmetry property: $F(u,v,w) = F(u,w,v).$

= Central triangles of Type 1 =

Let {{tmath|f(u,v,w)}} and {{tmath|g(u,v,w)}} be two triangle center functions, not both identically zero functions, having the same degree of homogeneity. Let {{mvar|a, b, c}} be the side lengths of the reference triangle {{math|△ABC}}. An {{math|(f, g)}}-central triangle of Type 1 is a triangle {{math|△A'B'C' }} the trilinear coordinates of whose vertices have the following form:{{cite web |last1=Weisstein, Eric W |title=Central Triangle |url=https://mathworld.wolfram.com/CentralTriangle.html |website=MathWorld--A Wolfram Web Resource. |publisher=MathWorld |access-date=17 December 2021}}{{cite journal |last1=Kimberling, C |title=Triangle Centers and Central Triangles |journal=Congressus Numerantium. A Conference Journal on Numerical Themes. 129 |date=1998 |volume=129}}{{bcn|date=May 2024}}

$\begin{array}{rcccccc}
A' =& f(a,b,c) &:& g(b,c,a) &:& g(c,a,b) \\
B' =& g(a,b,c) &:& f(b,c,a) &:& g(c,a,b) \\
C' =& g(a,b,c) &:& g(b,c,a) &:& f(c,a,b)
\end{array}$

= Central triangles of Type 2 =

Let {{tmath|f(u,v,w)}} be a triangle center function and {{tmath|g(u,v,w)}} be a function function satisfying the homogeneity property and having the same degree of homogeneity as {{tmath|f(u,v,w)}} but not satisfying the bisymmetry property. An {{math|(f, g)}}-central triangle of Type 2 is a triangle {{math|△A'B'C' }} the trilinear coordinates of whose vertices have the following form:{{bcn|date=May 2024}}

$\begin{array}{rcccccc}
A' =& f(a,b,c) &:& g(b,c,a) &:& g(c,b,a) \\
B' =& g(a,c,b) &:& f(b,c,a) &:& g(c,a,b) \\
C' =& g(a,b,c) &:& g(b,a,c) &:& f(c,a,b)
\end{array}$

= Central triangles of Type 3 =

Let {{tmath|g(u,v,w)}} be a triangle center function. An {{mvar|g}}-central triangle of Type 3 is a triangle {{math|△A'B'C' }} the trilinear coordinates of whose vertices have the following form:{{bcn|date=May 2024}}

$\begin{array}{rrcrcr}
A' =& 0 \quad\ \ &:& g(b,c,a) &:& - g(c,b,a) \\
B' =& - g(a,c,b) &:& 0 \quad\ \ &:& g(c,a,b) \\
C' =& g(a,b,c) &:& - g(b,a,c) &:& 0 \quad\ \
\end{array}$

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

Special cases

If {{math|1=f = g}}, the {{math|(f, g)}}-central triangle of Type 1 degenerates to the triangle center {{mvar|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 {{math|△ABC}} is a central triangle of Type 1. This is obtained by taking $f(u,v,w) = -1,\ g(u,v,w) = 1.$

Let {{mvar|X}} be a triangle center defined by the triangle center function {{tmath|g(a,b,c).}} Then the cevian triangle of {{mvar|X}} is a {{math|(0, g)}}-central triangle of Type 1.{{cite web |last1=Weisstein, Eric W |title=Cevian Triangle |url=https://mathworld.wolfram.com/CevianTriangle.html |website=MathWorld--A Wolfram Web Resource. |publisher=MathWorld |access-date=18 December 2021}}{{bcn|date=May 2024}}

Let {{mvar|X}} be a triangle center defined by the triangle center function {{tmath|f(a,b,c).}} Then the anticevian triangle of {{mvar|X}} is a {{math|(−f, f)}}-central triangle of Type 1.{{cite web |last1=Weisstein, Eric W |title=Anticevian Triangle |url=https://mathworld.wolfram.com/AnticevianTriangle.html |website=MathWorld--A Wolfram Web Resource. |publisher=MathWorld |access-date=18 December 2021}}{{bcn|date=May 2024}}

The Lucas central triangle is the {{math|(f, g)}}-central triangle with

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

S_A = \tfrac{1}{2}(b^2 + c^2 - a^2).

{{cite web |last1=Weisstein, Eric W |title=Lucas Central Triangle |url=https://mathworld.wolfram.com/LucasCentralTriangle.html |website=MathWorld--A Wolfram Web Resource. |publisher=MathWorld |access-date=18 December 2021}}{{bcn|date=May 2024}}

= Type 2 =

Let {{mvar|X}} be a triangle center. The pedal and antipedal triangles of {{mvar|X}} are central triangles of Type 2.{{cite web |last1=Weisstein, Eric W |title=Pedal Triangle |url=https://mathworld.wolfram.com/PedalTriangle.html |website=MathWorld--A Wolfram Web Resource. |publisher=MathWorld |access-date=18 December 2021}}{{bcn|date=May 2024}}
Yff Central Triangle{{cite web |last1=Weisstein, Eric W |title=Yff Central Triangle |url=https://mathworld.wolfram.com/YffCentralTriangle.html |website=MathWorld--A Wolfram Web Resource. |publisher=MathWorld |access-date=18 December 2021}}{{bcn|date=May 2024}}

References

Category:Objects defined for a triangle