Antiparallelogram

It can be formed from an isosceles trapezoid by adding the two diagonals and removing two parallel sides.

An antiparallelogram is a special case of a crossed quadrilateral, with two pairs of equal-length edges.

[4] Every antiparallelogram is a cyclic quadrilateral, meaning that its four vertices all lie on a single circle.

[3] Additionally, the four extended sides of any antiparallelogram are the bitangents of two circles, making antiparallelograms closely related to the tangential quadrilaterals, ex-tangential quadrilaterals, and kites (which are both tangential and ex-tangential).

[9] One form of a non-uniform but flexible polyhedron, the Bricard octahedron, can be constructed as a bipyramid over an antiparallelogram.

[2] A pair of nested antiparallelograms was used in a linkage defined by Alfred Kempe as part of Kempe's universality theorem, stating that any algebraic curve may be traced out by the joints of a suitably defined linkage.

Kempe called the nested-antiparallelogram linkage a "multiplicator", as it could be used to multiply an angle by an integer.

As the linkage moves, each antiparallelogram formed can be divided into two congruent triangles meeting at the crossing point.

In the triangle based on the fixed edge, the lengths of the two moving sides sum to the constant length of one of the antiparallelogram's crossed edges, and therefore the moving crossing point traces out an ellipse with the fixed points as its foci.

Symmetrically, the second (moving) uncrossed edge of the antiparallelogram has as its endpoints the foci of a second ellipse, formed from the first one by reflection across a tangent line through the crossing point.

[19] In the n-body problem, the study of the motions of point masses under Newton's law of universal gravitation, an important role is played by central configurations, solutions to the n-body problem in which all of the bodies rotate around some central point as if they were rigidly connected to each other.

For four bodies, with two pairs of the bodies having equal masses (but with the ratio between the masses of the two pairs varying continuously), numerical evidence indicates that there exists a continuous family of central configurations, related to each other by the motion of an antiparallelogram linkage.