09-07-2026, 10:49 PM
Summary
An antiparallelogram (also called a crossed parallelogram or contraparallelogram) is a self-intersecting quadrilateral in which the two pairs of opposite sides have equal lengths. Unlike an ordinary parallelogram, the corresponding sides are not parallel and one pair crosses the other. The figure has an axis of symmetry and its four vertices lie on the same circle, so it is a cyclic quadrilateral. It may also be constructed from an isosceles trapezoid by replacing its two parallel sides with its diagonals.
A notable property is that its signed area is zero, because it consists of two congruent triangular regions with opposite orientations. Its ordinary geometric area, however, is nonzero. If the relevant side lengths are $p$ and $q$ and their separation is $h$, the area can be written as
$A=\frac{hpq}{p+q}$.
Another interesting property is that the midpoints of all four sides are collinear. Thus, the Varignon parallelogram that normally arises by joining the midpoints of a quadrilateral degenerates into a single line segment.
Antiparallelograms also occur in mechanical linkages, particularly four-bar linkages known as butterfly or bow-tie linkages. Their motion can generate curves such as ellipses and, in special cases, the Bernoulli lemniscate. They therefore have applications in mechanical systems, non-circular gears, flexible polyhedra, and even configurations studied in celestial mechanics.
ARTICLE
An antiparallelogram (also called a crossed parallelogram or contraparallelogram) is a self-intersecting quadrilateral in which the two pairs of opposite sides have equal lengths. Unlike an ordinary parallelogram, the corresponding sides are not parallel and one pair crosses the other. The figure has an axis of symmetry and its four vertices lie on the same circle, so it is a cyclic quadrilateral. It may also be constructed from an isosceles trapezoid by replacing its two parallel sides with its diagonals.
A notable property is that its signed area is zero, because it consists of two congruent triangular regions with opposite orientations. Its ordinary geometric area, however, is nonzero. If the relevant side lengths are $p$ and $q$ and their separation is $h$, the area can be written as
$A=\frac{hpq}{p+q}$.
Another interesting property is that the midpoints of all four sides are collinear. Thus, the Varignon parallelogram that normally arises by joining the midpoints of a quadrilateral degenerates into a single line segment.
Antiparallelograms also occur in mechanical linkages, particularly four-bar linkages known as butterfly or bow-tie linkages. Their motion can generate curves such as ellipses and, in special cases, the Bernoulli lemniscate. They therefore have applications in mechanical systems, non-circular gears, flexible polyhedra, and even configurations studied in celestial mechanics.
ARTICLE
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