Editing Pascal's theorem (section)

==Proofs==

Pascal's original note<ref name= orig/> has no proof, but there are various modern proofs of 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. This was realised by Pascal, whose first lemma states the theorem for a circle. His second lemma states that what is true in one plane remains true upon projection to another plane.<ref name= orig/> 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 {{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.

A short proof can be constructed using cross-ratio preservation. Projecting tetrad {{math|''ABCE''}} from {{math|''D''}} onto line {{math|''AB''}}, we obtain tetrad {{math|''ABPX''}}, and projecting tetrad {{math|''ABCE''}} from {{math|''F''}} onto line {{math|''BC''}}, we obtain tetrad {{math|''QBCY''}}. This therefore means that {{math|''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, {{math|''XYZ''}} are collinear.

Another proof for Pascal's theorem for a circle uses [[Menelaus' theorem]] repeatedly.

[[Germinal Pierre Dandelin|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.

There also exists a simple proof for Pascal's theorem for a circle using the [[law of sines]] and [[Similarity (geometry)|similarity]].