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单词 PappussTheorem
释义

Pappus’s theorem


Let A,B,C be points on a line (not necessarily in that order) and let D,E,F points on another line (not necessarily in that order). Then the intersectionMathworldPlanetmath points of AD with FC, DB with CE, and BF with EA, are collinearMathworldPlanetmath.

This is a special case of Pascal’s mystic hexagram.

Remark. Pappus’s theorem is a statement about the incidence relationPlanetmathPlanetmath between points and lines in any geometric structureMathworldPlanetmath with points, lines, and an incidence relation between the points and the lines. Generally speaking, an incidence geometry is Pappian or satisfies the Pappian property if the statement of Pappus’s theorem is true. In both Euclidean and affine geometryMathworldPlanetmath, Pappus theorem is true. In plane projective geometryMathworldPlanetmath, both Pappian and non-Pappian planes exist. Furthermore, it can be shown that every Pappian plane is Desarguesian, and the converseMathworldPlanetmath is true if the plane is finite (the result of Wedderburn’s theorem).

TitlePappus’s theorem
Canonical namePappussTheorem
Date of creation2013-03-22 12:25:01
Last modified on2013-03-22 12:25:01
Ownerdrini (3)
Last modified bydrini (3)
Numerical id9
Authordrini (3)
Entry typeTheorem
Classificationmsc 51A05
SynonymPappus Theorem
Related topicPascalsMysticHexagram
Related topicCollinear
Related topicConcurrentMathworldPlanetmath
DefinesPappian
DefinesPappian property
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更新时间:2025/5/4 12:19:00