Triangle Center - Formal Definition

Formal Definition

A real-valued function f of three real variables a, b, c may have the following properties:

• Homogeneity: f(ta,tb,tc) = tn f(a,b,c) for some constant n and for all t > 0.
• Bisymmetry in the second and third variables: f(a,b,c) = f(a,c,b).

If a non-zero f has both these properties it is called a triangle center function. If f is a triangle center function and a, b, c are the side-lengths of a reference triangle then the point whose trilinear coordinates are f(a,b,c) : f(b,c,a) : f(c,a,b) is called a triangle center.

This definition ensures that triangle centers of similar triangles meet the invariance criteria specified above. By convention only the first of the three trilinear coordinates of a triangle center is quoted since the other two are obtained by cyclic permutation of a, b, c. This process is known as cyclicity.

Every triangle center function corresponds to a unique triangle center. This correspondence is not bijective. Different functions may define the same triangle center. For example the functions f₁(a,b,c) = 1/a and f₂(a,b,c) = bc both correspond to the centroid. Two triangle center functions define the same triangle center if and only if their ratio is a function symmetric in a, b and c.

Even if a triangle center function is well-defined everywhere the same cannot always be said for its associated triangle center. For example let f(a, b, c) be 0 if a/b and a/c are both rational and 1 otherwise. Then for any triangle with integer sides the associated triangle center evaluates to 0:0:0 which is undefined.