Magdalena LAMPA BACZYŃSKA, Daniel WÓJCIK

Relative conics and their Brianchon points

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.

PDF

___

1] Agarwal M, Natarajan N. Inheritance relations of hexagons and ellipses. The College Mathematics Journal 2016; 47 (3): 208-214. doi: 10.4169/college.math.j.47.3.208
[2] Baralić D, Spasojević I. Illumination of Pascal’s hexagrammum and octagrammum mysticum. Discrete and Computational Geometry 2015; 53: 414-427. doi: 10.1007/s00454-014-9658-6
[3] Conway J, Ryba A. The Pascal mysticum demystified. The Mathematical Intelligencer 2012; 34: 4-8. doi: 10.1007/s00283-012-9301-4
[4] Glaeser G, Stachel H, Odehnal B. The Universe of Conics: From the ancient Greeks to 21st century developments. Berlin Heidelberg: Springer Spektrum, 2016.
[5] Lord E. Symmetry and Pattern in Projective Geometry. London Heidelberg New York Dordrecht: Springer, 2013.
[6] Pokora P, Tutaj-Gasińska H. Harbourne constants and conic configurations on the projective plane. Mathematische Nachrichten 2016; 289 (7): 888-894. doi: 10.1002/mana.201500253
[7] Rigby JF. Configurations of circles and points. Journal of the London Mathematical Society 1983; s2-28 (1): 131-148. doi: 10.1112/jlms/s2-28.1.131
[8] Rigby JF. On the Money-Coutts configuration of nine anti-tangent cycles. Proceedings of the London Mathematical Society 1981; 3 (43): 110-132. doi: 10.1112/plms/s3-43.1.110