# 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 |