The term

Pascal's theorem is valid more generally for conics, even for degenerate such. F.e. two intersecting (or parallel) lines can be considered to form a degenerate conic. In this case Pascal's theorem is identical with the theorem of Pappus on hexagons

At first glance the theorem seems to be formidable and need a complicated proof argument. It is though a simple consequence of the properties of cross ratios (see CrossRatio0.html ).

The cross ratios (FEJH) and (MEDK) are equal, hence lines FM, DJ and KH are concurrent. This simple proof may be transferred verbatim to the more general case of an hexagon inscribed in a conic. All the ingredients of the proof are valid in this more general setting.

The cross ratios (FEJH) and (MEDK) are equal because they are cut off on lines e, g by the line bundles C(FEDB) and A(FEDB). But the four points (FEDB) as well as A, C being on a circle(conic) the cross ratios of these line bundles are equal (see CrossRatio.html ).

Pascal's theorem has several consequences and particular cases, examined in the references given below. It has also an obvious converse (called Maclaurin-Braikenridge theorem).

CrossRatio.html

CrossRatio0.html

CrossRatioLines.html

Duality.html

GoodParametrization.html

Harmonic.html

Harmonic_Bundle.html

HomographicRelation.html

HomographicRelationExample.html

PappusLines.html

Pascal2.html

PascalImmortel.html

PascalOnQuadrangles.html

PascalOnTriangles.html

ProjectivityFixingVertices.html

TriangleCircumconics.html

TriangleCircumconics2.html

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