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All the propositions in Projective Geometry occur in dual pairs which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words ``point'' and ``line.'' A similar duality exists for Reciprocation (Casey 1893).
See also Brianchon's Theorem, Conservation of Number Principle, Desargues' Theorem, Dual Polyhedron, Pappus's Hexagon Theorem, Pascal's Theorem, Permanence of Mathematical Relations Principle, Projective Geometry, Reciprocation
References
Casey, J. ``Theory of Duality and Reciprocal Polars.'' Ch. 13 in
A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing
an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis,
& Co., pp. 382-392, 1893.
Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 107-110, 1990.