Triangle Centers as Functions

Major Centers of Triangles

“…Then P is called (in [6] and [8]) a tnangle center, or simply a center. It is easy to veriiFy that the five special points already mentioned are centers.…”

Editorial board of advisors

“…For i, j = 1, 2, 3, let M ij be a midcircle of the circles 23 and M 31 are collinear. Since the radical center P of the triad C i , i = 1, 2, 3, has the same power with respect to these circles, they form a pencil and their common points X and Y are the poles of inversion mapping the circles C 1 , C 2 and C 3 into congruent circles.…”

“…By a non-degenerate triangle ABC, we mean an ordered triple (A, B, C) of non-collinear points in a fixed Euclidean plane E. Non-degenerate triangles form a subset of E 3 that we denote by T. For a subset U of T, the set of triples (a, b, c) ∈ R 3 that occur as the side-lengths of a triangle in U is denoted by U 0 . Thus In the spirit of [23] - [25], a symmetric triangle center function (or simply, a center function, or a center) is defined as a function that assigns to every triangle in T (or more generally in some subset U of T) a point in its plane in a manner that is symmetric and that respects isometries and dilations. Writing Z(A, B, C) as a barycentric combination of the position vectors A, B, and C, and letting a, b, and c denote the side-lengths of ABC in the standard order, we see that a center function Z on U is of the form…”

Editorial board of advisors

“…If we choose ε ij to satisfy ε 12 · ε 23 · ε 31 = −1, then the centers of M 12 , M 23 and M 31 are collinear. Since the radical center P of the triad C i , i = 1, 2, 3, has the same power with respect to these circles, they form a pencil and their common points X and Y are the poles of inversion mapping the circles C 1 , C 2 and C 3 into congruent circles.…”