Pascal's theorem

Pascal's theorem states that, for any hexagon inscribed in a conic, the three intersections of pairs of opposite sides are collinear. The line containing them is called the Pascal line.

Conic section: A curve formed by intersecting a plane with a cone, or described algebraically by a second-degree equation. Pascal's theorem applies to points on any nondegenerate conic.
Pascal line: The line containing the three intersections of opposite sides of an inscribed hexagon. It is the line whose existence Pascal's theorem guarantees.
Blaise Pascal: A French mathematician, physicist, and philosopher whose work ranged from geometry and probability to fluid mechanics. He discovered the hexagon theorem as a teenager and published it in 1640.
Hexagrammum Mysticum: The configuration of lines and intersection points generated by six points on a conic. Pascal's theorem is the first central collinearity in this rich configuration.
Projective plane: A geometric plane extended to include points at infinity, where parallel lines meet. The theorem's intersections remain defined when opposite sides are parallel.
Bézout's theorem: A theorem stating that plane algebraic curves intersect in a number of points determined by their degrees, counted with multiplicity. Algebraic methods use intersections of conics and lines to establish Pascal-type incidences.
Essai pour les coniques: Blaise Pascal's 1640 essay announcing results about conic sections, including his hexagon theorem. This is the work in which Pascal stated the theorem.
Steiner's theorem: A theorem describing repeated Pascal lines generated by six points on a conic. It identifies a systematic pattern among the many Pascal lines of one hexagram.
Collinearity: The property of points lying on the same straight line. The theorem's conclusion is precisely the collinearity of three side intersections.
Brianchon theorem: A projective theorem stating that the three main diagonals of a hexagon circumscribed about a conic are concurrent. It is the dual incidence statement obtained by exchanging points and lines.

Linked from 14 pages