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Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid: It has two adjacent angles that are supplementary, that is, they add up to 180 degrees. The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal.
Equivalently, it is a quadrilateral whose four sides can be grouped into two pairs of adjacent equal-length sides. [1] [7] A kite can be constructed from the centers and crossing points of any two intersecting circles. [8] Kites as described here may be either convex or concave, although some sources restrict kite to mean only convex kites.
Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite. [5] However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms ...
A right kite with its circumcircle and incircle. The leftmost and rightmost vertices have right angles. In Euclidean geometry, a right kite is a kite (a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other) that can be inscribed in a circle. [1]
A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed quadrilateral, butterfly quadrilateral or bow-tie quadrilateral. In a crossed quadrilateral, the four "interior" angles on either side of the crossing (two acute and two reflex , all on the left or all on the right as the figure is traced out) add up to 720°.
The base pairs form a parallelogram with half the area of the quadrilateral, A q, as the sum of the areas of the four large triangles, A l is 2 A q (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, A s is a quarter of A l (half linear dimensions yields quarter area), and the area of the parallelogram is A ...
Conversely, any quadrilateral in which a 2 + c 2 = b 2 + d 2 must be orthodiagonal. [5] This can be proved in a number of ways, including using the law of cosines, vectors, an indirect proof, and complex numbers. [6] The diagonals of a convex quadrilateral are perpendicular if and only if the two bimedians have equal length. [6]
A star p/q-trapezohedron (where 2 ≤ q < 1p) is defined by a regular zig-zag skew star 2p/q-gon base, two symmetric apices with no degree of freedom right above and right below the base, and quadrilateral faces connecting each pair of adjacent basal edges to one apex.