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- Pappuss_hexagon_theorem abstract "In mathematics, Pappus' 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. Pascal's theorem is in turn a special case of Cayley–Bacharach_theorem.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.".
- Pappuss_hexagon_theorem thumbnail Pappusconfig.svg?width=300.
- Pappuss_hexagon_theorem wikiPageExternalLink PappusDual.shtml.
- Pappuss_hexagon_theorem wikiPageExternalLink Pappus.shtml.
- Pappuss_hexagon_theorem wikiPageID "1493395".
- Pappuss_hexagon_theorem wikiPageLength "12639".
- Pappuss_hexagon_theorem wikiPageOutDegree "44".
- Pappuss_hexagon_theorem wikiPageRevisionID "672129799".
- Pappuss_hexagon_theorem wikiPageWikiLink Bipartite_graph.
- Pappuss_hexagon_theorem wikiPageWikiLink Category:Articles_containing_proofs.
- Pappuss_hexagon_theorem wikiPageWikiLink Category:Euclidean_plane_geometry.
- Pappuss_hexagon_theorem wikiPageWikiLink Category:Projective_geometry.
- Pappuss_hexagon_theorem wikiPageWikiLink Category:Theorems_in_geometry.
- Pappuss_hexagon_theorem wikiPageWikiLink Category:Theorems_in_plane_geometry.
- Pappuss_hexagon_theorem wikiPageWikiLink Cayley–Bacharach_theorem.
- Pappuss_hexagon_theorem wikiPageWikiLink Collinear.
- Pappuss_hexagon_theorem wikiPageWikiLink Collinearity.
- Pappuss_hexagon_theorem wikiPageWikiLink Concurrent_lines.
- Pappuss_hexagon_theorem wikiPageWikiLink Configuration_(geometry).
- Pappuss_hexagon_theorem wikiPageWikiLink Cross-ratio.
- Pappuss_hexagon_theorem wikiPageWikiLink Cross_ratio.
- Pappuss_hexagon_theorem wikiPageWikiLink Cut-the-Knot.
- Pappuss_hexagon_theorem wikiPageWikiLink Cut-the-knot.
- Pappuss_hexagon_theorem wikiPageWikiLink Degenerate_conic.
- Pappuss_hexagon_theorem wikiPageWikiLink Desargues_theorem.
- Pappuss_hexagon_theorem wikiPageWikiLink Desarguesian.
- Pappuss_hexagon_theorem wikiPageWikiLink Desarguess_theorem.
- Pappuss_hexagon_theorem wikiPageWikiLink Distance-regular_graph.
- Pappuss_hexagon_theorem wikiPageWikiLink Division_ring.
- Pappuss_hexagon_theorem wikiPageWikiLink Duality_(projective_geometry).
- Pappuss_hexagon_theorem wikiPageWikiLink Euclid.
- Pappuss_hexagon_theorem wikiPageWikiLink Gerhard_Hessenberg.
- Pappuss_hexagon_theorem wikiPageWikiLink Intersection_theorem.
- Pappuss_hexagon_theorem wikiPageWikiLink John_Wiley_&_Sons.
- Pappuss_hexagon_theorem wikiPageWikiLink Levi_graph.
- Pappuss_hexagon_theorem wikiPageWikiLink Limiting_case_(mathematics).
- Pappuss_hexagon_theorem wikiPageWikiLink Line_(geometry).
- Pappuss_hexagon_theorem wikiPageWikiLink Line_(mathematics).
- Pappuss_hexagon_theorem wikiPageWikiLink Non-Desarguesian_plane.
- Pappuss_hexagon_theorem wikiPageWikiLink Pappus_configuration.
- Pappuss_hexagon_theorem wikiPageWikiLink Pappus_graph.
- Pappuss_hexagon_theorem wikiPageWikiLink Pappus_of_Alexandria.
- Pappuss_hexagon_theorem wikiPageWikiLink Pascals_theorem.
- Pappuss_hexagon_theorem wikiPageWikiLink Permanent.
- Pappuss_hexagon_theorem wikiPageWikiLink Perspective_(geometry).
- Pappuss_hexagon_theorem wikiPageWikiLink Projective_configuration.
- Pappuss_hexagon_theorem wikiPageWikiLink Projective_plane.
- Pappuss_hexagon_theorem wikiPageWikiLink Special_case.
- Pappuss_hexagon_theorem wikiPageWikiLink File:Pappus-collection-7-129.svg.
- Pappuss_hexagon_theorem wikiPageWikiLink File:Pappus-collection-7-136.svg.
- Pappuss_hexagon_theorem wikiPageWikiLink File:Pappus-collection-7-137.svg.
- Pappuss_hexagon_theorem wikiPageWikiLink File:Pappus-collection-7-138.svg.
- Pappuss_hexagon_theorem wikiPageWikiLink File:Pappus_hexagon.svg.
- Pappuss_hexagon_theorem wikiPageWikiLink File:Pappusconfig.svg.
- Pappuss_hexagon_theorem wikiPageWikiLinkText "Pappus line".
- Pappuss_hexagon_theorem wikiPageWikiLinkText "Pappus' concept".
- Pappuss_hexagon_theorem wikiPageWikiLinkText "Pappus's (hexagon) theorem".
- Pappuss_hexagon_theorem wikiPageWikiLinkText "Pappus's Theorem".
- Pappuss_hexagon_theorem wikiPageWikiLinkText "Pappus's hexagon theorem".
- Pappuss_hexagon_theorem wikiPageWikiLinkText "Pappus's theorem".
- Pappuss_hexagon_theorem wikiPageWikiLinkText "Theorem of Pappus".
- Pappuss_hexagon_theorem wikiPageWikiLinkText "hexagon theorem".
- Pappuss_hexagon_theorem wikiPageWikiLinkText "pappian plane".
- Pappuss_hexagon_theorem wikiPageWikiLinkText "pappian".
- Pappuss_hexagon_theorem hasPhotoCollection Pappuss_hexagon_theorem.
- Pappuss_hexagon_theorem wikiPageUsesTemplate Template:Citation.
- Pappuss_hexagon_theorem wikiPageUsesTemplate Template:Reflist.
- Pappuss_hexagon_theorem subject Category:Articles_containing_proofs.
- Pappuss_hexagon_theorem subject Category:Euclidean_plane_geometry.
- Pappuss_hexagon_theorem subject Category:Projective_geometry.
- Pappuss_hexagon_theorem subject Category:Theorems_in_geometry.
- Pappuss_hexagon_theorem subject Category:Theorems_in_plane_geometry.
- Pappuss_hexagon_theorem hypernym Collinear.
- Pappuss_hexagon_theorem comment "In mathematics, Pappus' 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.".
- Pappuss_hexagon_theorem label "Pappus's hexagon theorem".
- Pappuss_hexagon_theorem sameAs Pappova_věta.
- Pappuss_hexagon_theorem sameAs Satz_von_Pappos.
- Pappuss_hexagon_theorem sameAs Teorema_del_hexágono_de_Pappus.
- Pappuss_hexagon_theorem sameAs Pappoksen_lause.
- Pappuss_hexagon_theorem sameAs Théorème_de_Pappus.
- Pappuss_hexagon_theorem sameAs Papposz-tétel.
- Pappuss_hexagon_theorem sameAs Teorema_dellesagono_di_Pappo.
- Pappuss_hexagon_theorem sameAs Stelling_van_Pappos.
- Pappuss_hexagon_theorem sameAs Twierdzenie_Pappusa.
- Pappuss_hexagon_theorem sameAs Teorema_ëd_Pap.
- Pappuss_hexagon_theorem sameAs Teorema_de_Pappus.
- Pappuss_hexagon_theorem sameAs m.055ptl.
- Pappuss_hexagon_theorem sameAs Teorema_lui_Pappus.
- Pappuss_hexagon_theorem sameAs Теорема_Паппа.
- Pappuss_hexagon_theorem sameAs Pappos_sats.
- Pappuss_hexagon_theorem sameAs ทฤษฎีบทของปัปปุส.
- Pappuss_hexagon_theorem sameAs Теорема_Паппа.
- Pappuss_hexagon_theorem sameAs Định_lý_Pappus_(6_điểm).
- Pappuss_hexagon_theorem sameAs Q851166.
- Pappuss_hexagon_theorem sameAs Q851166.
- Pappuss_hexagon_theorem sameAs 帕普斯定理.
- Pappuss_hexagon_theorem wasDerivedFrom Pappuss_hexagon_theoremoldid=672129799.
- Pappuss_hexagon_theorem depiction Pappusconfig.svg.
- Pappuss_hexagon_theorem isPrimaryTopicOf Pappuss_hexagon_theorem.