crossed quadrilateral

A complete crossed quadrilateral is formed by four distinct lines AC, AD, CF and DE in the Euclidean planeMathworldPlanetmath, 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 segmentsMathworldPlanetmath CE, CF, DE and DF. Its diagonalsMathworldPlanetmath CD and EF are outside of the crossed quadrilateral. In the picture below, the same quadrilateralMathworldPlanetmath as above is still in cyan, and its diagonals are drawn in blue.


The sum of the inner angles of CEDF is 720o. Its area is obtained e.g. ( by of the Bretschneider’s formulaMathworldPlanetmathPlanetmath (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 bisectorMathworldPlanetmath 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 productPlanetmathPlanetmath of the both diagonals CD and EF (yellow in the diagram) has a value, of the inner angles (e.g. on α).

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