Generalization of Kimberling’s Concept of Triangle Center for Other Polygons

$n$-gon centers and central lines

“…The main contributions herein are the following. First, in Section 2, we provide a new formal definition of center of a n-gon, which is more clear than the one given in [15]. We prove that our definition and the one in [15] are equivalent (Theorem 2.3) and, as a consequence, we solve one of the open questions in that article.…”

“…First, in Section 2, we provide a new formal definition of center of a n-gon, which is more clear than the one given in [15]. We prove that our definition and the one in [15] are equivalent (Theorem 2.3) and, as a consequence, we solve one of the open questions in that article. Then, after revisiting the ideas of central points and lines for trigons in Section 3, we study possible generalizations of the concept of central line for n ≥ 4.…”

“…For this section the main reference is [15], but the notation, the use of some terms and the approach are slightly different in this paper. In [15] the authors first introduce n-gon center functions and then define centers, which are a geometric interpretation of the first concept.…”

“…Finally in [15] the authors succesfully generalized the definition of triangle (trigon) center function given by Kimberling to n-gons for n ≥ 4.…”

“…We show that some desirable properties for central lines that hold for trigons (see Remark 3.3) do not hold for central lines of n-gons, for n ≥ 4 (see Theorem 5.4 and Example 5.5). In Section 6 we study the problem that was already mentioned in [1,5,15] concerning the relationship between regularity and coincidence of centers. We obtain a result in this direction (see Theorem 6.3).…”