Quadrilateral
A four-sided Polygon sometimes (but not very often) also known as a Tetragon. If not explicitly stated, all fourVertices are generally taken to lie in a Plane. If the points do not lie in a Plane, thequadrilateral is called a Skew Quadrilateral.
For a planar convex quadrilateral (left figure above), let the lengths of the sides be 
, 
, 
, and 
, theSemiperimeter 
, and the Diagonals 
and 
. The Diagonals are Perpendicular Iff 
. An equation for the sum of the squares of side lengths is
 | (1) |
is the length of the line joining the Midpoints of the Diagonals.The Area of a quadrilateral is given by |  |  | (2) |
|  | (3) | |
|  | (4) | |
|  | (5) |
where (4) is known as Bretschneider's Formula (Beyer 1987).
A special type of quadrilateral is the Cyclic Quadrilateral, for which a Circle can be circumscribed so that ittouches each Vertex. For Bicentric quadrilaterals, theCircumcircle and Incircle satisfy
 | (6) |
is the Circumradius, 
in the Inradius, and is the separation of centers. A quadrilateralwith two sides Parallel is called a Trapezoid.There is a relationship between the six distances 
, 
, 
, 
, 
, and 
between the four points of a quadrilateral (Weinberg 1972):
 | |  | |
 |  | ||
|  | ||
|  | ||
|  | ||
|  | ||
|  | (7) |
See also Bimedian, Brahmagupta's Formula, Bretschneider's Formula, Complete Quadrilateral,Cyclic-Inscriptable Quadrilateral, Cyclic Quadrilateral, Diamond, Eight-Point CircleTheorem, Equilic Quadrilateral, Fano's Axiom, Léon Anne's Theorem,Lozenge, Orthocentric Quadrilateral, Parallelogram, Ptolemy's Theorem, RationalQuadrilateral, Rhombus, Skew Quadrilateral, Trapezoid, Varignon's Theorem, von Aubel'sTheorem, Wittenbauer's Parallelogram
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987. Routh, E. J. ``Moment of Inertia of a Quadrilateral.'' Quart. J. Pure Appl. Math. 11, 109-110, 1871. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 7, 1972.
- Hilbert Function
- Hilbert Hotel
- Hilbert Matrix
- Hilbert Number
- Hilbert Polynomial
- Hilbert's Axioms
- Hilbert-Schmidt Norm
- Hilbert-Schmidt Theory
- Hilbert's Constants
- Hilbert's Inequality
- Hilbert's Nullstellansatz
- Hilbert Space
- Hilbert's Problems
- Hilbert's Theorem
- Hilbert Transform
- Hillam's Theorem
- Hill Determinant
- Hill's Differential Equation
- Hindu Check
- Hinge
- Hippias' Quadratrix
- Hippopede
- Hi-Q
- Histogram
- Hitch