Triangle centers as functions (Q1328237)

In the Euclidean plane \(\mathbb{R}^ 2\) let be given a triangle \(\Delta A_ 1 A_ 2 A_ 3\) with respective sidelengths \(a_ 1, a_ 2, a_ 3\). By `homogeneous trilinear coordinates' \(x_ 1, x_ 2, x_ 3\) of a point \(X \in \mathbb{R}^ 2\) one understands any triple of real numbers proportional to the directed distances from \(X\) to the sides \(A_ 2 A_ 3\), \(A_ 3 A_ 1\), \(A_ 1 A_ 2\), respectively, of the reference triangle \(\Delta A_ 1 A_ 2 A_ 3\). A point \(Y(y_ 1 : y_ 2 : y_ 3)\) is called a center of \(\Delta A_ 1 A_ 2 A_ 3\) iff there exists a function \(f(x_ 1, x_ 2, x_ 3)\) in \(\mathbb{R}\) which is (F2) symmetric with respect to \(x_ 2, x_ 3\) and (F3) homogeneous of some degree \(n \in \mathbb{N}_ 0\) such that (F1) \(y_ 1 : y_ 2 : y_ 3 = f(a_ 1, a_ 2, a_ 3) : f(a_ 2, a_ 3, a_ 1) : f(a_ 3, a_ 1, a_ 2)\). Examples for such centers are the centroid \((a_ 2a_ 3 : a_ 3a_ 1 : a_ 1a_ 2)\) -- which apparently belongs to \(f(x_ 1,x_ 2,x_ 3) = x_ 2x_ 3\) or, equivalently, to \(f(x_ 1,x_ 2,x_ 3) = 1/x_ 1\) --, the incenter, circumcenter and orthocenter of \(\Delta A_ 1 A_ 2 A_ 3\). A center can be understood as the `value' \(f(a_ 1, a_ 2, a_ 3)\): \(f(a_ 2, a_ 3, a_ 1)\): \(f(a_ 3, a_ 1, a_ 2)\) of an appropriate function \(f\) on the set of all triangles \(\Delta A_ 1 A_ 2 A_ 3\) (each of them being represented by \(a_ 1,a_ 2,a_ 3)\): then, problems about centers lead to functional equations involving \(f\). This idea is applied to several kinds of problems, among them the two- triangle problem \(A_ 1XX_ 1\): For any center \(X\) -- being the value of \(f\) in \(\Delta\)- of a variable triangle \(\Delta = \Delta A_ 1 A_ 2 A_ 3\) let \(X_ 1\) be the value of \(f\) in \(\Delta XA_ 2A_ 3\); for which choices of the center \(X\) (that is, of \(f)\) are the points \(A_ 1,X,X_ 1\) collinear? The author presents an equivalent functional equation for \(f\); among the well-known triangle-centers, only centroid and orthocenter have been found to satisfy the problem.