A synthetic proof of A. Myakishev's generalization of van Lamoen circle theorem and an application (Q2825787)

The von Lamoen circle theorem runs as follows: ``If a triangle is divided by the three medians into six smaller triangles, then the circumcenters of these smaller triangles lie on a circle''.NEWLINENEWLINESome proofs of that theorem to be found in [\textit{M. H. Nguyen}, Forum Geom. 5, 127--132 (2005; Zbl 1119.51010); \textit{F. van Lamoen}, ``Circumcenters on a circle: 10830'', Am. Math. Mon. 109, No. 4, 396--397 (2002; \url{doi:10.2307/2695516}); \textit{K. Y. Li}, ``Concyclic problems'', Math. Excalibur 6, No. 1 (2001), \url{https://www.math.ust.hk/excalibur/v6_n1.pdf}; \textit{A. Myakishev} and \textit{P. Y. Woo}, Forum Geom. 3, 57--63 (2003; Zbl 1028.51027)]. In Hyacinthos message 5612 (2002), \textit{A. Myakishev} (in collaboration with \textit{B. Wolk}) gave a generalization of van Lamoen's theorem. (The formulation of that theorem is reprinted in the introduction of the paper under review).NEWLINENEWLINEIn the paper under review, the author gives an alternative proof of Myakishev's theorem thereby providing that there exists a family of six circumcenters lying on a circle relative to a triangle associated with Kiepert's configuration.NEWLINENEWLINE The mathematics in the paper is straightforward.NEWLINENEWLINE The English language as written by the author, contains many lapses, grammatical mistakes and the like. The editors and/or referee of the paper should have been paid serious attention about these things.