\(n\)-gon centers and central lines (Q6590661)

Classical triangle centers such as circumcenter, incenter, orthocenter play a crucial role in plane geometry. The central lines (i.e., the lines that connects two centers) such as the Euler line, the Nagel line, also hold significant importance.\N\NAs noted by the authors, \textit{C. Kimberling} presented an abstract definition of triangle centers in [Aequationes Math. 45, No. 2--3, 127--152 (1993; Zbl 0774.39006); Math. Mag. 67, No. 3, 163--187 (1994; Zbl 0821.51014)]. Subsequently, in [the second author and \textit{R. Sánchez-Cauce}, Result. Math. 76, No. 2, Paper No. 81, 18 p. (2021; Zbl 1467.51013)] efforts were made to generalize Kimberling's work in the framework of \(n\)-gons. \N\NIn the paper under review, these ideas are further developed and some results of the aforementioned paper are complemented. Roughly speaking, these centers are described using \emph{barycentric-like} functions of vertices \((V_1,\ldots,V_n)\) of a given \(n\)-gon \(P\), along with the distance matrix \((d_{ij})_{n\times n}\) of \(P\) where \(d_{ij}=d(V_i,V_j)\). Furthermore, several open problems and applications related to the subject are discussed.