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.
A triangle center function is a real valued function
of three real variables u, v, w having the following properties: Let
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][better source needed]
{\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}}}
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][better source needed]
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][better source needed]
This is a degenerate triangle in the sense that the points A', B', C' are collinear.
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.