Pappus's hexagon theorem

In a projective plane, the three intersections of opposite sides of a hexagon whose vertices alternate between two lines are collinear.

Pappus configuration: A configuration of nine points and nine lines in which each point lies on three lines and each line contains three points. The theorem's six vertices and three intersection points form this incidence pattern.
Projective plane: A geometric structure in which any two distinct points determine one line and any two distinct lines meet at one point. This is the setting that guarantees every pair of opposite sides has an intersection.
Pappus of Alexandria: A Greek mathematician of late antiquity whose Collection preserves and extends results from earlier geometry. The theorem bears his name through its appearance in the geometric tradition associated with him.
Pappian plane: A projective plane in which every configuration satisfying Pappus's hypotheses also satisfies its collinearity conclusion. The theorem's validity throughout a plane defines this important class of projective planes.
Desargues' theorem: A projective geometry theorem stating that corresponding intersections of two perspective triangles are collinear. It offers a related incidence theorem that can help explain Pappian collinearity.
Projective geometry: Geometry that studies properties unchanged by projective transformations, including incidence and cross-ratio. Pappus's theorem is a foundational statement about incidence in this geometry.
Pappus's Collection: A late-antique Greek mathematical work assembling and commenting on results from earlier authors. Its geometric material is linked to the historical source of the theorem's name.
Commutative field: A field whose multiplication is commutative, such as the real or complex numbers. Projective planes over fields satisfy Pappus's theorem, reflecting commutative multiplication.
Projective coordinates: A coordinate system representing points by nonzero vectors up to nonzero scalar multiplication. Homogeneous coordinates turn the theorem's incidences into algebraic equations.
Collinearity: The property of three or more points lying on one straight line. Collinearity is the precise conclusion asserted for the three side-intersection points.

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