geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a central triangle is a triangle in the plane of the reference triangle. The

trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...

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 Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size, ...

. At least one of the two functions must be a triangle center function. The

excentral triangle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter ...

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 of three real variables having the following properties: :* Homogeneity property:

F(tu,tv,tw) = t^n F(u,v,w)

for some constant and for all . The constant is the degree of homogeneity of the function :*Bisymmetry property:

F(u,v,w) = F(u,w,v).

Central triangles of Type 1

Let and be two triangle center functions, not both identically zero functions, having the same degree of homogeneity. Let be the side lengths of the reference triangle . An -central triangle of Type 1 is a triangle the trilinear coordinates of whose vertices have the following form:

\begin
  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

Central triangles of Type 2

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 -central triangle of Type 2 is a triangle the trilinear coordinates of whose vertices have the following form:

\begin
  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

Central triangles of Type 3

Let be a triangle center function. An -central triangle of Type 3 is a triangle the trilinear coordinates of whose vertices have the following form:

\begin
  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

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

Special cases

If , the -central triangle of Type 1 degenerates to the triangle center . All central triangles of both Type 1 and Type 2 relative to an equilateral triangle degenerate to a point.

Examples

Type 1

*The

of triangle is a central triangle of Type 1. This is obtained by taking

f(u,v,w) = -1,\ g(u,v,w) = 1.

*Let be a triangle center defined by the triangle center function Then the cevian triangle of is a -central triangle of Type 1. *Let be a triangle center defined by the triangle center function Then the anticevian triangle of is a -central triangle of Type 1. *The Lucas central triangle is the -central triangle with

f(a,b,c) = a(2S+S_2), \quad g(a,b,c) = aS_A,

where is twice the area of triangle ABC and

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

Type 2

*Let be a triangle center. The

pedal A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to: Computers and other equipment * Footmouse, a foot-operated computer mouse * In medical transcription, a pedal is used to control p ...

and antipedal triangles of are central triangles of Type 2. * Yff Central Triangle

References

{{reflist Objects defined for a triangle