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Line 41: A short elementary proof of Pascal's theorem in the case of a circle was found by {{harvtxt\|van Yzeren\|1993}}, based on the proof in {{harv\|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 {{harvtxt\|Stefanovic\|2010}}. We can infer the proof from existence of [[isogonal conjugate]] too. If we are to show that {{math\|''X'' {{=}} ''AB'' ∩ ''DE''}}, {{math\|''Y'' {{=}} ''BC'' ∩ ''EF''}}, {{math\|''Z'' {{=}} ''CD'' ∩ ''FA''}} are collinear for concyclic {{math\|''ABCDEF''}}, then notice that {{math\|△''EYB''}} and {{math\|△''CYF''}} are similar, and that {{math\|''X''}} and {{math\|''Z''}} will correspond to the isogonal conjugate if we overlap the similar triangles. This means that {{math\|∠''CYX'' {{=}} ∠''CYZ''}}, hence making {{math\|''XYZ''}} collinear.

Pascal's theorem: Difference between revisions