# Menelaus’ theorem

If the points $X$, $Y$ and $Z$ are on the sides of a triangle $ABC$ (including their prolongations), collinear^{} and do not coincide with any of the points $A$, $B$ and $C$, then the equation

$$\frac{AZ}{ZB}\cdot \frac{BY}{YC}\cdot \frac{CX}{XA}=-1$$ |

holds (all segments are directed line segments). The converse of this theorem also holds (thus: three points on the prolongations of the triangle’s sides are collinear if the above equation holds).

Title | Menelaus’ theorem |
---|---|

Canonical name | MenelausTheorem |

Date of creation | 2013-03-22 12:46:20 |

Last modified on | 2013-03-22 12:46:20 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 7 |

Author | mathwizard (128) |

Entry type | Theorem |

Classification | msc 51A05 |

Related topic | CevasTheorem |

Related topic | TrigonometricVersionOfCevasTheorem |

Related topic | Collinear |