Quasi-circumcenters and a generalization of the quasi-Euler line to a hexagon (Q2878856)

Let \(ABCD\) be a plane quadrilateral that is not cyclic. In his paper in [Forum Geom. 6, 289--295 (2006; Zbl 1129.51012)], \textit{A. Myakishev} defines the quasi-circumcenter \(\mathcal{O}\) of \(ABCD\) to be the intersection of the lines \(\mathcal{O}_A \mathcal{O}_C\) and \(\mathcal{O}_B\mathcal{O}_D\), where \(\mathcal{O}_U\) is the circumcenter of the triangle obtained from \(ABCD\) by deleting vertex \(U\). Defining the quasi-orthocenter \(\mathcal{H}\) of \(ABCD\) in a similar fashion, and denoting the lamina centroid of \(ABCD\) by \(\mathcal{G}\), Myakishev then proves that \(\mathcal{O}\), \(\mathcal{H}\), and \(\mathcal{G}\) are collinear with \(\|\mathcal{O} \mathcal{H}\| : \|\mathcal{H} \mathcal{G}\| = 3 : -2\). This is a satisfactory Euler-like theorem for arbitrary quadrilaterals.NEWLINENEWLINENEWLINENEWLINEThe author of the paper under review defines the quasi-circumcenter of an arbitrary hexagon \(A_1A_2A_3A_4A_5A_6\) as follows: He lets \(P_i\) be the Myakishev quasi-circumcenter of quadrilateral \(A_i A_{i+1}A_{i+2}A_{i+3}\), where indices are significant only up to their values mod 6, he proves that the three main diagonals of hexagon \(P_1P_2P_3P_4P_5P_6\) are concurrent, and he calls the point \(\mathcal{O}\) of concurrence the quasi-circumcenter of hexagon \(A_1A_2A_3A_4A_5A_6\). Defining the quasi-orthocenter \(\mathcal{H}\) in the same way, and denoting the lamina centroid by \(\mathcal{G}\), the author proves that \(\mathcal{O}\), \(\mathcal{H}\) are collinear with \(\|\mathcal{O} \mathcal{H}\| : \|\mathcal{H} \mathcal{G}\| = 3 : -2\), thus proving a satisfactory Euler-like theorem for arbitrary hexagon. He shows that the same thing cannot be done for octagons.NEWLINENEWLINENEWLINENEWLINEThis reviewer would like to add that another type of quasi-circumcenter was defined for quadrilaterals by \textit{O. Radko} and \textit{E. Tsukerman} [Forum Geom. 12, 161--189 (2012; Zbl 1244.51006)]. Here one starts with a non-cyclic quadrilateral \(Q_0\), and inductively defines \(Q_{n+1}\), \(n \geq 0\), to be the quadrilateral enclosed among the perpendicular bisectors of the sides of \(Q_n\), and then proves that these quadrilaterals shrink to a point that may be called the Tsukerman quasi-circumcenter of \(Q\). It would be interesting to see how the quasi-circumcenters of Tsukerman and Myakishev are related.