Brianchon's theorem

Brianchon's theorem states that the three lines joining opposite vertices of a hexagon circumscribed about a conic meet at one point, called the Brianchon point.

Conic section: A curve formed by intersecting a plane with a cone, including ellipses, parabolas, and hyperbolas. The theorem applies to hexagons whose six sides are tangent to one such curve.
Pascal's theorem: A theorem stating that opposite sides of a hexagon inscribed in a conic meet in three collinear points. It is the projective dual of Brianchon's concurrency statement.
Charles Julien Brianchon: A French mathematician whose work included the theorem on circumscribed hexagons now bearing his name. Brianchon published the result in 1806 while studying conic sections.
Brianchon point: The common point of the three lines joining opposite vertices of a circumscribed hexagon. It is the specific point whose existence the theorem guarantees.
Projective geometry: The study of geometric properties preserved by projection, including incidence and cross-ratio. Its incidence language explains concurrency without requiring distances or angles.
Projective duality: A correspondence in projective geometry that exchanges points with lines and preserves incidence relations. It turns Pascal's collinearity for an inscribed hexagon into Brianchon's concurrence for a circumscribed one.
Projective geometry in the nineteenth century: The development of geometry centered on projection, incidence, and transformations during the nineteenth century. Brianchon's theorem emerged from this era's systematic study of conics and duality.
Complete quadrilateral: The configuration formed by four lines, their six pairwise intersections, and the associated opposite pairs. Its incidence relations help organize constructions involving conics and hexagons.
Tangency: The condition in which curves or surfaces meet with a shared tangent at a point. Each side of the hexagon touches the conic at a tangent point.
Pole and polar: A point-line correspondence defined by a conic in projective geometry. Conic polarity provides a geometric framework for exchanging the point and line configurations.

Linked from 9 pages