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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. i have food
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.
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