Pascal's theorem: Difference between revisions

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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]] (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 at three points which lie on a straight line, called the '''Pascal line''' of the hexagon. It is named after [[Blaise Pascal]].

Pascal's theorem is a very useful theorem in Olympiad geometry to prove the collinearity of three intersections among six points on a circle.

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.