$n$-gon centers and central lines

Abstract

In this paper we provide a review of the concept of center of a n-gon, generalizing the original idea given by C. Kimberling for triangles. We also generalize the concept of central line for n-gons for n ≥ 3 and establish its basic properties.

The concept of center as an equivariant map and a proof of an analogue of the center conjecture for equifacetal simplices

“…, P k such that all the elements are different and two consecutive vertex are adjacent. Some examples of polygon centers may be found in [4,12,13]. In this case, that corresponds to the first abuse of notation explained in the remark above, two different polygons may have the same vertices and a given center may assign different points to them.…”

“…Many authors have followed an approximation, similar to the one by Kimberling, in other contexts. For instance, we can find a definition of n-gons center in [4,12,13] and a similar one of center of a set of n-points in [3,12]. So, in Section 2 of the present paper, we push the concept of center to its maximum generality in order to provide a general formal definition in terms of equivariant maps between G-spaces.…”

“…(b) Let S be the group of congruences that fix V . Then for every point P ∈ R 2 fixed by S, there exists a triangle center Z : A → R 2 such that Z(V ) = P (see [4] for a proof of an equivalent statement).…”