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# 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 reduced 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. by means 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 symmetric 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 constant value, independent of the inner angles (e.g. on $\alpha$).

## Mathematics Subject Classification

51-00*no label found*

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