In this paper, we study the parameter spaces for Böröczky arrangements Bn of n lines, where n < 12. We prove that up to n = 12, there exist only one arrangement nonrealizable over the rational numbers, that is B 11 .
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.
The purpose of this paper is to study the famous Pappus configuration of 9 lines and its dual arrangement. We show among others that by applying the Pappus Theorem to the dual arrangement we obtain the configuration corresponding to the initial data of beginning configuration. We consider also the Pappus arrangements with some additional incidences and we establish algebraic conditions paralleling with these incidences.
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