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