A purely synthetic proof of Dao's theorem on six circumcenters associated with a cyclic hexagon
Let \(A_1A_2A_3A_4A_5A_6\) be a hexagon, and let the subscripts \(A_i\) be taken modulo 6. For \(1 \leq i \leq 6\), let \(B_{i+3}\) be the point where \(A_iA_{i+1}\) and \(A_{i+2}A_{i+3}\) intersect, and let \(G_{i+3}\) be the circumcenter of \(A_iA_{i+1}B_{i+2}\). In his recent paper in [Forum Geom. 14, 243--246 (2014; Zbl 1301.51017)], \textit{N. Dergiades} proves, using complex numbers, that if the hexagon is cyclic, then the lines \(G_1G_4\), \(G_2G_5\), and \(G_3G_6\) are concurrent, and he attributes this theorem to T. A. Dao. In the paper under review, the author gives a new short synthetic proof of Dao's theorem that uses Ceva's theorem and its converse.
scientific article; zbMATH DE number 6353597