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)
|
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)
|
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)
|
 |  |  |
(4)
|
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)
|
 |  |  |
(6)
|
(Beyer 1987, p. 123), Bretschneider's formula
 |  |  |
(7)
|
 |  |  |
(8)
|
(Coolidge 1939; Ivanoff 1960; Beyer 1987, p. 123) where 
is the semiperimeter, and
the beautiful formula
![K=sqrt((s-a)(s-b)(s-c)(s-d)-abcdcos^2[1/2(A+B)])](/images/equations/Quadrilateral/NumberedEquation3.svg) |
(9)
|
(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)
|
This can be most simply derived by setting the left side of the Cayley-Menger determinant
 |
(11)
|
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.
