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).

Title Pappus’s theorem
Canonical name PappussTheorem
Date of creation 2013-03-22 12:25:01
Last modified on 2013-03-22 12:25:01
Owner drini (3)
Last modified by drini (3)
Numerical id 9
Author drini (3)
Entry type Theorem
Classification msc 51A05
Synonym Pappus Theorem
Related topic PascalsMysticHexagram
Related topic Collinear
Related topic ConcurrentMathworldPlanetmath
Defines Pappian
Defines Pappian property