A quadrilateral, sometimes also known as a tetragon or quadrangle (Johnson 1929, p. 61) is a four-sided polygon. If not explicitly stated, all four polygon vertices are generally taken to lie in a plane. (If the points do not lie in a plane, the quadrilateral is called a skew quadrilateral.) There are three topological types of quadrilaterals (Wenninger 1983, p. 50): convex quadrilaterals (left figure), concave quadrilaterals (middle figure), and crossed quadrilaterals (or butterflies, or bow-ties; right figure).
A quadrilateral with two sides parallel is called a trapezoid, whereas a quadrilateral with opposite pairs of sides parallel is called a parallelogram.
For a planar convex quadrilateral (left figure above), let the lengths of the sides be , , , and , the semiperimeter , and the polygon diagonals and . The polygon diagonals are perpendicular iff .
An equation for the sum of the squares of side lengths is
(1)
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where is the length of the line joining the midpoints of the polygon diagonals (Casey 1888, p. 22).
For bicentric quadrilaterals, the circumcircle and incircle satisfy
(2)
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where is the circumradius, in the inradius, and is the separation of centers.
Given any five points in the plane in general position, four will form a convex quadrilateral. This result is a special case of the so-called happy end problem (Hoffman 1998, pp. 74-78).
There is a beautiful formula for the area of a planar convex quadrilateral in terms of the vectors corresponding to its two diagonals. Represent the sides of the quadrilateral by the vectors , , , and arranged such that and the diagonals by the vectors and arranged so that and . Then
(3)
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(4)
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where is the determinant and is a two-dimensional cross product.
There are a number of beautiful formulas for the area of a planar convex quadrilateral in terms of the side and diagonal lengths, including
(5)
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(6)
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(Beyer 1987, p. 123), Bretschneider's formula
(7)
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(8)
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(Coolidge 1939; Ivanoff 1960; Beyer 1987, p. 123) where is the semiperimeter, and the beautiful formula
(9)
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(Bretschneider 1842; Strehlke 1842; Coolidge 1939; Beyer 1987, p. 123).
The centroid of the vertices of a quadrilateral occurs at the point of intersection of the bimedians (i.e., the lines and joining pairs of opposite midpoints) (Honsberger 1995, pp. 36-37). In addition, it is the midpoint of the line connecting the midpoints of the diagonals and (Honsberger 1995, pp. 39-40).
The four angle bisectors of a quadrilateral intersect adjacent bisectors in four concyclic points (Honsberger 1995, p. 35).
Any non-self-intersecting quadrilateral tiles the plane.
There is a relationship between the six distances , , , , , and between the four points of a quadrilateral (Weinberg 1972):
(10)
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This can be most simply derived by setting the left side of the Cayley-Menger determinant
(11)
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equal to 0 (corresponding to a tetrahedron of volume 0), thus giving a relationship between the distances between vertices of a planar quadrilateral (Uspensky 1948, p. 256).
A special type of quadrilateral is the cyclic quadrilateral, for which a circle can be circumscribed so that it touches each polygon vertex. Another special type is a tangential quadrilateral, for which a circle and be inscribed so it is tangent to each edge. A quadrilateral that is both cyclic and tangential is called a bicentric quadrilateral.