Desargues's theorem

In a projective plane, if lines joining corresponding vertices of two triangles meet at one point, the intersections of corresponding sides are collinear.

Perspective triangles: Two triangles are perspective from a point when lines joining corresponding vertices are concurrent. This is the hypothesis: corresponding vertex-joining lines pass through one point.
Projective geometry: The study of geometric properties preserved by projective transformations, especially incidence and cross-ratio. Desargues's theorem is a foundational incidence result in this geometry.
Girard Desargues: A seventeenth-century French mathematician and engineer whose work helped establish projective geometry. The theorem bears his name and appeared in his work on conic sections.
Converse of Desargues's theorem: The result that collinear intersections of corresponding sides imply concurrency of corresponding vertex-joining lines. It reverses the theorem's implication for the same pair of triangles.
Desargues axis: The line containing the three intersections of corresponding sides of two perspective triangles. It is the collinear line guaranteed by the theorem.
Collinearity: The property of points lying on a single line. The conclusion says the three intersections of corresponding sides are collinear.
Blaise Pascal: A French mathematician whose theorem on a hexagon inscribed in a conic became a major result in projective geometry. Pascal's theorem belongs to the projective tradition shaped by Desargues's work.
Desargues configuration: A configuration of ten points and ten lines encoding the incidences in Desargues's theorem. It collects the theorem's vertices, centers, axis, and side intersections into one incidence pattern.
Desargues center: The common point of the three lines joining corresponding vertices of perspective triangles. It is the concurrency point required by the theorem's hypothesis.
Concurrency: The property of multiple lines passing through one common point. The premise says the three lines joining corresponding vertices are concurrent.

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