Jaydeep Chipalkatti, On the Coincidences of Pascal Lines,
Forum Geometricorum, 16 (2016) 1--21.

Abstract. Let K denote a smooth conic in the complex projective plane. Pascal's theorem says that, given six points A, B, C, D, E, F on K, the three intersection points AE \cap BF, AD \cap CF, BD \cap CE are collinear. This defines the Pascal line of the array

 A B C F E D

and one gets sixty such lines in general by permuting the points. In this paper we consider the variety ψ of sextuples {A,…, F}, for which some of the Pascal lines coincide. We show that ψ has two irreducible components: a five-dimensional component of sextuples in involution, and a four-dimensional component of what will be called `ricochet configurations'. This gives a complete synthetic characterization of points in ψ. The proof relies upon Gröbner basis techniques to solve multivariate polynomial equations; the implementation was done in two distinct computer algebra systems.

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