Twins of conic hexagons (Q6623809)

Six pairwise distinct points \(P_1,P_2,\dots ,P_6\) on a non-degenerate conic \(C\), called a conic hexa-set, define 15 lines which in turn yield, in general, 45 intersection points different from the initial points. The intersection of the lines \(P_iP_j\) and \(P_kP_l\) is denoted by \(P_{ijkl}\). Let \(S\) be the set of these 45 points \(P_{ijkl}\). A hexagon with vertices in \(S\) is called a Pascal twin of the original hexagon with vertices \(P_i\). A hexagon with \(k\geq 1\) vertices among the points \(P_1,P_2,\dots ,P_6\) and \(6-k\) vertices in \(S\) is called a Siamese Pascal twin of the original hexagon with vertices \(P_i\). In this paper, the authors study the Pascal twins and the Siamese Pascal twins of a conic hexagon.