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- Pappus's_hexagon_theorem abstract "In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon AbCaBc. It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem" is valid are called pappian planes. The dual of this incidence theorem states that given one set of concurrent lines A, B, C, and another set of concurrent lines a, b, c, then the lines x, y, z defined by pairs of points resulting from pairs of intersections A∩b and a∩B, A∩c and a∩C, B∩c and b∩C are concurrent. (Concurrent means that the lines pass through one point.)Pappus's theorem is a special case of Pascal's theorem for a conic—the limiting case when the conic degenerates into 2 straight lines.The Pappus configuration is the configuration of 9 lines and 9 points that occurs in Pappus's theorem, with each line meeting 3 of the points and each point meeting 3 lines. In general, the Pappus line does not pass through the point of intersection of ABC and abc. This configuration is self dual. Since, in particular, the lines Bc, bC, XY have the properties of the lines x, y, z of the dual theorem, and collinearity of X, Y, Z is equivalent to concurrence of Bc, bC, XY, the dual theorem is therefore just the same as the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges.".
- Pappus's_hexagon_theorem thumbnail Pappusconfig.svg?width=300.
- Pappus's_hexagon_theorem wikiPageExternalLink PappusDual.shtml.
- Pappus's_hexagon_theorem wikiPageExternalLink Pappus.shtml.
- Pappus's_hexagon_theorem wikiPageID "1493395".
- Pappus's_hexagon_theorem wikiPageRevisionID "592828973".
- Pappus's_hexagon_theorem hasPhotoCollection Pappus's_hexagon_theorem.
- Pappus's_hexagon_theorem subject Category:Articles_containing_proofs.
- Pappus's_hexagon_theorem subject Category:Projective_geometry.
- Pappus's_hexagon_theorem subject Category:Theorems_in_geometry.
- Pappus's_hexagon_theorem type Abstraction100002137.
- Pappus's_hexagon_theorem type Communication100033020.
- Pappus's_hexagon_theorem type Message106598915.
- Pappus's_hexagon_theorem type Proposition106750804.
- Pappus's_hexagon_theorem type Statement106722453.
- Pappus's_hexagon_theorem type Theorem106752293.
- Pappus's_hexagon_theorem type TheoremsInGeometry.
- Pappus's_hexagon_theorem comment "In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon AbCaBc.".
- Pappus's_hexagon_theorem label "Pappus's hexagon theorem".
- Pappus's_hexagon_theorem label "Satz von Pappos".
- Pappus's_hexagon_theorem label "Stelling van Pappos".
- Pappus's_hexagon_theorem label "Teorema de Pappus".
- Pappus's_hexagon_theorem label "Teorema del hexágono de Pappus".
- Pappus's_hexagon_theorem label "Teorema dell'esagono di Pappo".
- Pappus's_hexagon_theorem label "Théorème de Pappus".
- Pappus's_hexagon_theorem label "Twierdzenie Pappusa".
- Pappus's_hexagon_theorem label "Теорема Паппа".
- Pappus's_hexagon_theorem label "帕普斯定理".
- Pappus's_hexagon_theorem sameAs Pappova_věta.
- Pappus's_hexagon_theorem sameAs Satz_von_Pappos.
- Pappus's_hexagon_theorem sameAs Teorema_del_hexágono_de_Pappus.
- Pappus's_hexagon_theorem sameAs Théorème_de_Pappus.
- Pappus's_hexagon_theorem sameAs Teorema_dell'esagono_di_Pappo.
- Pappus's_hexagon_theorem sameAs Stelling_van_Pappos.
- Pappus's_hexagon_theorem sameAs Twierdzenie_Pappusa.
- Pappus's_hexagon_theorem sameAs Teorema_de_Pappus.
- Pappus's_hexagon_theorem sameAs m.055ptl.
- Pappus's_hexagon_theorem sameAs Q851166.
- Pappus's_hexagon_theorem sameAs Q851166.
- Pappus's_hexagon_theorem sameAs Pappus's_hexagon_theorem.
- Pappus's_hexagon_theorem wasDerivedFrom Pappus's_hexagon_theorem?oldid=592828973.
- Pappus's_hexagon_theorem depiction Pappusconfig.svg.
- Pappus's_hexagon_theorem isPrimaryTopicOf Pappus's_hexagon_theorem.