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Quadrilateral  

Six different types of quadrilaterals  
Edges and vertices  4 
Schläfli symbol  {4} (for square) 
Area  various methods; see below 
Internal angle (degrees)  90° (for square and rectangle) 
Quadrilateral  

Six different types of quadrilaterals  
Edges and vertices  4 
Schläfli symbol  {4} (for square) 
Area  various methods; see below 
Internal angle (degrees)  90° (for square and rectangle) 
In Euclidean plane geometry, a quadrilateral is a polygon with four sides (or edges) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5sided), hexagon (6sided) and so on.
The origin of the word "quadrilateral" is the two Latin words quadri, a variant of four, and latus, meaning "side".
Quadrilaterals are simple (not selfintersecting) or complex (selfintersecting), also called crossed. Simple quadrilaterals are either convex or concave.
The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is
This is a special case of the ngon interior angle sum formula (n − 2) × 180°. In a crossed quadrilateral, the four interior angles on either side of the crossing add up to 720°.^{[1]}
All convex quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.
A parallelogram is a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms also include the square, rectangle, rhombus and rhomboid.
The two diagonals of a convex quadrilateral are the line segments that connect opposite vertices.
The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides.^{[8]} They intersect at the "vertex centroid" of the quadrilateral (see Remarkable points below).
The four maltitudes of a convex quadrilateral are the perpendiculars to a side through the midpoint of the opposite side.^{[9]} .
There are various general formulas for the area K of a convex quadrilateral.
The area can be expressed in trigonometric terms as
where the lengths of the diagonals are p and q and the angle between them is θ.^{[10]} In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to since θ is 90°.
Bretschneider's formula^{[11]} expresses the area in terms of the sides and two opposite angles:
where the sides in sequence are a, b, c, d, where s is the semiperimeter, and A and C are two (in fact, any two) opposite angles. This reduces to Brahmagupta's formula for the area of a cyclic quadrilateral when A+C = 180°.
Another area formula in terms of the sides and angles, with angle C being between sides b and c, and A being between sides a and d, is
In the case of a cyclic quadrilateral, the latter formula becomes
In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to
Alternatively, we can write the area in terms of the sides and the intersection angle θ of the diagonals, so long as this angle is not 90°:^{[12]}
In the case of a parallelogram, the latter formula becomes
Another area formula including the sides a, b, c, d is^{[13]}
where x is the distance between the midpoints of the diagonals and φ is the angle between the bimedians.
The following two formulas expresses the area in terms of the sides a, b, c, d, the semiperimeter s, and the diagonals p, q:
The first reduces to Brahmagupta's formula in the cyclic quadrilateral case, since then pq = ac + bd.
The area can also be expressed in terms of the bimedians m, n and the diagonals p, q:
The area of a quadrilateral ABCD can be calculated using vectors. Let vectors AC and BD form the diagonals from A to C and from B to D. The area of the quadrilateral is then
which is half the magnitude of the cross product of vectors AC and BD. In twodimensional Euclidean space, expressing vector AC as a free vector in Cartesian space equal to (x_{1},y_{1}) and BD as (x_{2},y_{2}), this can be rewritten as:
If a convex quadrilateral has the consecutive sides a, b, c, d and the diagonals p, q, then its area K satisfies^{[18]}
From Bretschneider's formula it directly follows that the area of a quadrilateral satisfies
with equality if and only if the quadrilateral is cyclic or degenerate such that one side is equal to the sum of the other three (it has collapsed into a line segment, so the area is zero).
The area of any quadrilateral also satisfies the inequality^{[19]}
In the following table it is listed if the diagonals in some of the most basic quadrilaterals bisect each other, if their diagonals are perpendicular, and if their diagonals have equal length.^{[20]} The list applies to the most general cases, and excludes named subsets.
Quadrilateral  Bisecting diagonals  Perpendicular diagonals  Equal diagonals 

Trapezoid  No  See note 1  No 
Isosceles trapezoid  No  See note 1  Yes 
Parallelogram  Yes  No  No 
Kite  See note 2  Yes  See note 2 
Rectangle  Yes  No  Yes 
Rhombus  Yes  Yes  No 
Square  Yes  Yes  Yes 
Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of (nonsimilar) trapezoids and isosceles trapezoids that do have perpendicular diagonals and are not any other named quadrilateral.
Note 2: In a kite, one diagonal bisects the other. The most general kite has unequal diagonals, but there is an infinite number of (nonsimilar) kites in which the diagonals are equal in length (and the kites are not any other named quadrilateral).
The length of the diagonals in a convex quadrilateral ABCD can be calculated using the law of cosines. Thus
and
Other, more symmetric formulas for the length of the diagonals, are^{[21]}
and
In any convex quadrilateral ABCD, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals plus four times the square of the line segment connecting the midpoints of the diagonals. Thus
where x is the distance between the midpoints of the diagonals.^{[22]}^{:p.126} This is sometimes known as Euler's quadrilateral theorem and is a generalization of the parallelogram law. A corollary is the inequality
where equality holds if and only if the quadrilateral is a parallelogram.
Euler also generalized Ptolemy's theorem, which is an equality in a cyclic quadrilateral, into an inequality for a convex quadrilateral. It states that
where there is equality if and only if the quadrilateral is cyclic.^{[22]}^{:p.128–129} This is often called Ptolemy's inequality.
The German mathematician Carl Anton Bretschneider derived in 1842 the following generalization of Ptolemy's theorem, regarding the product of the diagonals in a convex quadrilateral^{[23]}
This relation can be considered to be a law of cosines for a quadrilateral. In a cyclic quadrilateral, where A + C = 180°, it reduces to pq = ac + bd. Since cos (A + C) ≥ 1, it also gives a proof of Ptolemy's inequality.
If X and Y are the feet of the normals from B and D to the diagonal AC = p in a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD, d = DA, then^{[24]}^{:p.14}
In a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD, d = DA, and where the diagonals intersect at E,
where e = AE, f = BE, g = CE, and h = DE.^{[25]}
The shape of a convex quadrilateral is fully determined by the lengths of its sides in sequence and of one diagonal between two specified vertices. The two diagonals p, q and the four side lengths a, b, c, d of a quadrilateral are related^{[26]} by the CayleyMenger determinant, as follows:
The midpoints of the sides of any quadrilateral (convex, concave or crossed) are the vertices of a parallelogram called the Varignon parallelogram. It has the following properties:
The diagonals of the Varignon parallelogram are the bimedians of the original quadrilateral.
The two bimedians in a quadrilateral and the line segment joining the midpoints of the diagonals in that quadrilateral are concurrent and are all bisected by their point of intersection.^{[22]}^{:p.125}
In a convex quadrilateral with sides a, b, c and d, the length of the bimedian that connects the midpoints of the sides a and c is
where p and q are the length of the diagonals.^{[28]} The length of the bimedian that connects the midpoints of the sides b and d is
Hence^{[22]}^{:p.126}
This is also a corollary to the parallelogram law applied in the Varignon parallelogram.
The length of the bimedians can also be expressed in terms of two opposite sides and the distance x between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence^{[17]}
and
Note that the two opposite sides in these formulas are not the two that the bimedian connects.
In a convex quadrilateral, there are the following dual connection between the bimedians and the diagonals:^{[24]}
The four angles of a simple quadrilateral ABCD satisfy the following identities:^{[29]}
and
Also,^{[30]}
In the last two formulas, no angle is allowed to be a right angle, since then the tangent functions are not defined.
Among all quadrilaterals with a given perimeter, the one with the largest area is the square. This is called the isoperimetric theorem for quadrilaterals. It is a direct consequence of the area inequality^{[19]}^{:p.114}
where K is the area of a convex quadrilateral with perimeter L. Equality holds if and only if the quadrilateral is a square. The dual theorem states that of all quadrilaterals with a given area, the square has the shortest perimeter.
The quadrilateral with given side lengths that has the maximum area is the cyclic quadrilateral.^{[31]}
Of all convex quadrilaterals with given diagonals, the orthodiagonal quadrilateral has the largest area.^{[19]}^{:p.119} This is a direct consequence of the fact that the area of a convex quadrilateral satisfies
where θ is the angle between the diagonals p and q. Equality holds if and only if θ = 90°.
If P is an interior point in a convex quadrilateral ABCD, then
From this inequality it follows that the point inside a quadrilateral that minimizes the sum of distances to the vertices is the intersection of the diagonals. Hence that point is the Fermat point of a convex quadrilateral.^{[32]}^{:p.120}
The centre of a quadrilateral can be defined in several different ways. The "vertex centroid" comes from considering the quadrilateral as being empty but having equal masses at its vertices. The "side centroid" comes from considering the sides to have constant mass per unit length. The usual centre, called just centroid (centre of area) comes from considering the surface of the quadrilateral as having constant density. These three points are in general not all the same point.^{[33]}
The "vertex centroid" is the intersection of the two bimedians.^{[34]} As with any polygon, the x and y coordinates of the vertex centroid are the arithmetic means of the x and y coordinates of the vertices.
The "area centroid" of quadrilateral ABCD can be constructed in the following way. Let G_{a}, G_{b}, G_{c}, G_{d} be the centroids of triangles BCD, ACD, ABD, ABC respectively. Then the "area centroid" is the intersection of the lines G_{a}G_{c} and G_{b}G_{d}.^{[35]}
In a general convex quadrilateral ABCD, there are no natural analogies to the circumcenter and orthocenter of a triangle. But two such points can be constructed in the following way. Let O_{a}, O_{b}, O_{c}, O_{d} be the circumcenters of triangles BCD, ACD, ABD, ABC respectively; and denote by H_{a}, H_{b}, H_{c}, H_{d} the orthocenters in the same triangles. Then the intersection of the lines O_{a}O_{c} and O_{b}O_{d} is called the quasicircumcenter; and the intersection of the lines H_{a}H_{c} and H_{b}H_{d} is called the quasiorthocenter of the convex quadrilateral.^{[35]} These points can be used to define an Euler line of a quadrilateral. In a convex quadrilateral, the quasiorthocenter H, the "area centroid" G, and the quasicircumcenter O are collinear in this order, and HG = 2GO.^{[35]}
There can also be defined a quasininepoint center E as the intersection of the lines E_{a}E_{c} and E_{b}E_{d}, where E_{a}, E_{b}, E_{c}, E_{d} are the ninepoint centers of triangles BCD, ACD, ABD, ABC respectively. Then E is the midpoint of OH.^{[35]}
Another remarkable line in a convex quadrilateral is the Newton line.
A taxonomy of quadrilaterals is illustrated by the following graph. Lower forms are special cases of higher forms. Note that "trapezium" here is referring to the British definition (the North American equivalent is a trapezoid), and "kite" excludes the concave kite (arrowhead or dart). Inclusive definitions are used throughout.
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