Thomsen's theorem

Thomsen's theorem states that six successive perspectivities associated with a complete quadrilateral or triangle return a point on a line to its starting position, under the theorem's incidence conditions.

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Projectivity: A bijective projective transformation between projective lines that preserves cross-ratios. Each perspectivity in the six-step construction induces a projectivity between lines.
Projective geometry: The study of geometric properties preserved by projective transformations. Thomsen's theorem belongs to geometry that retains incidence while disregarding metric measurements.
Jakob Steiner: A Swiss mathematician known for foundational work in geometry, especially projective and synthetic geometry. The theorem is traditionally associated with Steiner's projective-geometric work.
Pascal's theorem: A theorem stating that opposite-side intersections of a hexagon inscribed in a conic are collinear. Both are projective closure results, but Pascal's theorem centers on a conic and six vertices.
Perspectivity: A projective correspondence between two lines induced by projecting from a fixed center. The theorem's successive mappings are perspectivities centered at triangle vertices.
Triangle: A polygon with three vertices and three sides. Its vertices and sides organize the sequence of perspectivities.
Karl Georg Christian von Staudt: A German mathematician who developed a synthetic foundation for projective geometry. His incidence-based approach clarifies the setting in which such closure theorems are stated.
Pappus's hexagon theorem: A projective theorem asserting collinearity of three intersection points formed from two triples of points on lines. It is another incidence theorem with a six-point configuration, but its conclusion is collinearity.
Complete quadrangle: A configuration of four points in a projective plane, together with the six lines joining pairs. Its incidence structure supplies the points and lines used in standard formulations of the closure.
Incidence geometry: The study of which geometric objects lie on or meet other objects. The theorem's statement and proof use only point-line incidence.