A complete crossed quadrilateral is formed by four distinct lines , , and in the Euclidean plane, each of which intersects the other three. The intersection of and is labelled as . A complete crossed quadrilateral has six vertices, of which and , and , and are opposite.
The complete crossed quadrilateral is often to the crossed quadrilateral (cyan in the diagram), consisting of the four line segments , , and . Its diagonals and 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 is . 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 and are equal; similarly, the lengths of the opposite sides and are equal. Below, an antiparallelogram is drawn in red. The antiparallelogram is with respect to the perpendicular bisector of the diagonal (which is also the perpendicular bisector of the diagonal ). When the lengths of the sides , , , and are fixed, the product of the both diagonals and (yellow in the diagram) has a value, of the inner angles (e.g. on ).
|Date of creation||2013-03-22 17:11:34|
|Last modified on||2013-03-22 17:11:34|
|Last modified by||pahio (2872)|
|Defines||complete crossed quadrilateral|