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 in \(A_i\) 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 \(A_iA_{i+1}B_{i+2}\). The author of the paper under review proves a theorem that he attributes to T. A. Dao and that states that if the hexagon is cyclic, then the lines \(G_1G_4\), \(G_2G_5\), and \(G_3G_6\) are concurrent. Although the converse is possibly too good to be true, one may wonder about what exactly the hexagons that have this property are. One may also ask whether the point of concurrence has a different and simpler description that does not resort to the ear triangles or to their circumcenters. The proof demonstrates the power of the algebra of complex numbers in handling problems in plane geometry.NEWLINENEWLINEThe afore-mentioned theorem of Dao seems to be new. At least it does not appear in the beautiful collection compiled by \textit{H. Walser} [99 points of intersection. Examples -- pictures -- proofs. Washington, DC: The Mathematical Association of America (MAA) (2006; Zbl 1112.00006)], where it would fit nicely alongside other points of intersection pertaining to hexagons, such as points 16, 17, 24, 58, and 60.
scientific article; zbMATH DE number 6349710