Proofs
Pascal's original statement and proof of the result differ from those ordinarily seen today, but there are various modern proofs for the theorem.
It is sufficient to prove the theorem when the conic is a circle, because any (non-degenerate) conic can be reduced to a circle by a projective transformation; degenerate conics follow by continuity (the theorem is true for non-degenerate conics, and thus holds in the limit of degenerate conic).
A short elementary proof of Pascal's theorem in the case of a circle was found by van Yzeren (1993), based on the proof in (Guggenheimer 1967). This proof proves the theorem for circle and then generalizes it to conics.
A short elementary computational proof in the case of the real projective plane was found by Stefanovic (2010)
We can infer the proof from existence of isogonal conjugate too. If we are to show that X = AB ∩ DE, Y = BC ∩ EF, Z = CD ∩ FA are collinear for conconical ABCDEF, then notice that ADY and CYF are similar, and that X and Z will correspond to the isogonal conjugate if we overlap the similar triangles. This means that angle DYX = angle CYZ, hence making XYZ collinear.
A short proof can be constructed using cross-ratio preservation. Projecting tetrad ABCE from D onto line AB, we obtain tetrad ABPX, and projecting tetrad ABCE from F onto line BC, we obtain tetrad QBCY. This therefore means that R(AB; PX) = R(QB; CY), where one of the points in the two tetrads overlap, hence meaning that other lines connecting the other three pairs must coincide to preserve cross ratio. Therefore XYZ are collinear.
Another proof for Pascal's theorem for a circle uses Menelaus' theorem repeatedly.
Dandelin, the geometer who discovered the celebrated Dandelin spheres, came up with a beautiful proof using "3D lifting" technique that is analogous to the 3D proof of Desargues' theorem. The proof makes use of the property that for every conic section we can find a one-sheet hyperboloid which passes through the conic. Also there exist a simple proof for Pascal's theorem for a circle uses Law of Sines and similarity.
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