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 intersection points of $AD$ with $FC$, $DB$ with $CE$, and $BF$ with $EA$, are collinear.

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

Remark. Pappus’s theorem is a statement about the incidence relation between points and lines in any geometric structure 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 geometry, Pappus theorem is true. In plane projective geometry, both Pappian and non-Pappian planes exist. Furthermore, it can be shown that every Pappian plane is Desarguesian, and the converse 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	Concurrent
Defines	Pappian
Defines	Pappian property