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







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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