Pascal's theorem: Difference between revisions

Revision as of 01:23, 19 December 2019

Revision as of 13:35, 23 January 2020

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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 conconicalconcyclic {{math|''ABCDEF''}}, then notice that {{math|△''ADYEYB''}} 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|∠''DYXBYX'' {{=}} ∠''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.