# crossed quadrilateral

A complete crossed quadrilateral is formed by four distinct lines $AC$, $AD$, $CF$ and $DE$ in the Euclidean plane^{}, each of which intersects the other three. The intersection of $CF$ and $DE$ is labelled as $B$. A complete crossed quadrilateral has six vertices, of which $A$ and $B$, $C$ and $D$, $E$ and $F$ are opposite.

The complete crossed quadrilateral is often to the crossed quadrilateral $CEDF$ (cyan in the diagram), consisting of the four line segments^{} $CE$, $CF$, $DE$ and $DF$. Its diagonals^{} $CD$ and $EF$ are outside of the crossed quadrilateral. In the picture below, the same quadrilateral^{} as above is still in cyan, and its diagonals are drawn in blue.

The sum of the inner angles of $CEDF$ is ${720}^{\mathrm{o}}$. Its area is obtained e.g. (http://planetmath.org/Eg) by of the Bretschneider’s formula^{} (cf. area of a quadrilateral).

A special case of the crossed quadrilateral is the antiparallelogram, in which the lengths of the opposite sides $CE$ and $DF$ are equal; similarly, the lengths of the opposite sides $CF$ and $DE$ are equal. Below, an antiparallelogram $CEDF$ is drawn in red. The antiparallelogram is with respect to the perpendicular bisector^{} of the diagonal $CD$ (which is also the perpendicular bisector of the diagonal $EF$). When the lengths of the sides $CE$, $CF$, $DE$, and $DF$ are fixed, the product^{} of the both diagonals $CD$ and $EF$ (yellow in the diagram) has a value, of the inner angles (e.g. on $\alpha $).

Title | crossed quadrilateral |
---|---|

Canonical name | CrossedQuadrilateral |

Date of creation | 2013-03-22 17:11:34 |

Last modified on | 2013-03-22 17:11:34 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 25 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 51-00 |

Related topic | PtolemysTheorem |

Defines | complete crossed quadrilateral |

Defines | antiparallelogram |