Relative conics and their Brianchon points

Abstract

The purpose of this paper is to study some additional relations between lines and points in the configuration of six lines tangent to the common conic. One of the most famous results concerning with this configuration is Brianchon theorem. It says that three diagonals of a hexagon circumscribing around conic are concurrent. They meet in the so called Brianchon point. In fact, by relabeling the vertices of hexagon, we obtain 60 distinct Brianchon points. We prove, among others, that, in the set of all intersection points of six tangents to the same conic, there exist exactly 10 sextuples of points lying on the common conic, which form the (106, 154) conic-point configuration. We establish a relation between all Brianchon points of these conics. We use both, algebraic and geometric tools.