Pascal's theorem: Difference between revisions

Revision as of 04:40, 30 July 2018

Revision as of 05:01, 30 July 2018

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[[Image:Pascal'sTheoremLetteredColored.PNG|thumb|250px|Self-crossing hexagon {{math|''ABCDEF''}}, inscribed in a circle. Its sides are extended so that pairs of opposite sides intersect on Pascal's line. Each pair of extended opposite sides has its own color: one red, one yellow, one blue. Pascal's line is shown in white.]]

In [[projective geometry]], '''Pascal's theorem''' (also known as the '''hexagrammum mysticum theorem''') states that if six arbitrary points are chosen on a [[conic section|conic]] (i.e.,which may be an [[ellipse]], [[parabola]] or [[hyperbola]] in an appropriate [[affine plane]]) and joined by line segments in any order to form a [[hexagon]], then the three pairs of opposite [[Edge (geometry)|sides]] of the hexagon ([[extended side|extended]] if necessary) meet in three points which lie on a straight line, called the '''Pascal line''' of the hexagon.

The theorem is also valid in the [[Euclidean plane]], but the statement needs to be adjusted to deal with the special cases when opposite sides are parallel.

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== Euclidean variants ==

The most natural setting for Pascal's theorem is in a [[projective plane]] since allany two lines meet and no exceptions need be made for parallel lines. However, the theorem remains valid in the Euclidean plane, with the correct interpretation of what happens when some opposite sides of the hexagon are parallel, the theorem remains valid in the Euclidean plane.

If exactly one pair of opposite sides of the hexagon are parallel, then the conclusion of the theorem is that the "Pascal line" determined by the two points of intersection is parallel to the parallel sides of the hexagon. If two pairs of opposite sides are parallel, then all three pairs of opposite sides form pairs of parallel lines and there is no Pascal line in the Euclidean plane (in this case, the [[line at infinity]] of the extended Euclidean plane is the Pascal line of the hexagon).