Quadrilateral

Last updated
Quadrilateral
Six Quadrilaterals.svg
Some 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 edges (sides) and four vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle), tetragon (in analogy to pentagon, 5-sided polygon, and hexagon, 6-sided polygon), and 4-gon (in analogy to k-gons for arbitrary values of k). A quadrilateral with vertices , , and is sometimes denoted as . [1] [2]

Contents

The word "quadrilateral" is derived from the Latin words quadri, a variant of four, and latus, meaning "side".

Quadrilaterals are either simple (not self-intersecting), or complex (self-intersecting, or 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 [2]

This is a special case of the n-gon interior angle sum formula: (n − 2) × 180°.

All non-self-crossing quadrilaterals tile the plane, by repeated rotation around the midpoints of their edges.

Simple quadrilaterals

Any quadrilateral that is not self-intersecting is a simple quadrilateral.

Convex quadrilaterals

Euler diagram of some types of simple quadrilaterals. (UK) denotes British English and (US) denotes American English. Euler diagram of quadrilateral types.svg
Euler diagram of some types of simple quadrilaterals. (UK) denotes British English and (US) denotes American English.
Convex quadrilaterals by symmetry, represented with a Hasse diagram. Symmetries of square.svg
Convex quadrilaterals by symmetry, represented with a Hasse diagram.

In a convex quadrilateral, all interior angles are less than 180°, and the two diagonals both lie inside the quadrilateral.

Quadrilaterals.svg

Concave quadrilaterals

In a concave quadrilateral, one interior angle is bigger than 180°, and one of the two diagonals lies outside the quadrilateral.

Complex quadrilaterals

An antiparallelogram DU21 facets.png
An antiparallelogram

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°. [9]

Special line segments

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. [11] They intersect at the "vertex centroid" of the quadrilateral (see § Remarkable points and lines in a convex quadrilateral below).

The four maltitudes of a convex quadrilateral are the perpendiculars to a side—through the midpoint of the opposite side. [12]

Area of a convex quadrilateral

There are various general formulas for the area K of a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD and d = DA.

Trigonometric formulas

The area can be expressed in trigonometric terms as [13]

where the lengths of the diagonals are p and q and the angle between them is θ. [14] In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to since θ is 90°.

The area can be also expressed in terms of bimedians as [15]

where the lengths of the bimedians are m and n and the angle between them is φ.

Bretschneider's formula [16] [13] 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, as long θ is not 90°: [17]

In the case of a parallelogram, the latter formula becomes

Another area formula including the sides a, b, c, d is [15]

where x is the distance between the midpoints of the diagonals, and φ is the angle between the bimedians.

The last trigonometric area formula including the sides a, b, c, d and the angle α (between a and b) is: [ citation needed ]

which can also be used for the area of a concave quadrilateral (having the concave part opposite to angle α), by just changing the first sign + to - .

Non-trigonometric formulas

The following two formulas express the area in terms of the sides a, b, c, d, the semiperimeter s, and the diagonals p, q:

[18]
[19]

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:

[20]
[21] :Thm. 7

In fact, any three of the four values m, n, p, and q suffice for determination of the area, since in any quadrilateral the four values are related by [22] :p. 126 The corresponding expressions are: [23]

if the lengths of two bimedians and one diagonal are given, and [23]

if the lengths of two diagonals and one bimedian are given.

Vector formulas

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 two-dimensional Euclidean space, expressing vector AC as a free vector in Cartesian space equal to (x1,y1) and BD as (x2,y2), this can be rewritten as:

Diagonals

Properties of the diagonals in some quadrilaterals

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. [24] The list applies to the most general cases, and excludes named subsets.

QuadrilateralBisecting diagonalsPerpendicular diagonalsEqual diagonals
Trapezoid NoSee note 1No
Isosceles trapezoid NoSee note 1Yes
Parallelogram YesNoNo
Kite See note 2YesSee note 2
Rectangle YesNoYes
Rhombus YesYesNo
Square YesYesYes

Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of (non-similar) 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 (non-similar) kites in which the diagonals are equal in length (and the kites are not any other named quadrilateral).

Lengths of the diagonals

The lengths of the diagonals in a convex quadrilateral ABCD can be calculated using the law of cosines on each triangle formed by one diagonal and two sides of the quadrilateral. Thus

and

Other, more symmetric formulas for the lengths of the diagonals, are [25]

and

Generalizations of the parallelogram law and Ptolemy's theorem

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.

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 [26]

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.

Other metric relations

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 [27] :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. [28]

The shape and size of a convex quadrilateral are 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 [13] by the Cayley-Menger determinant, as follows:

Angle bisectors

The internal angle bisectors of a convex quadrilateral either form a cyclic quadrilateral [22] :p.127 (that is, the four intersection points of adjacent angle bisectors are concyclic) or they are concurrent. In the latter case the quadrilateral is a tangential quadrilateral.

In quadrilateral ABCD, if the angle bisectors of A and C meet on diagonal BD, then the angle bisectors of B and D meet on diagonal AC. [29]

Bimedians

The Varignon parallelogram EFGH Varignon theorem convex.png
The Varignon parallelogram EFGH

The bimedians of a quadrilateral are the line segments connecting the midpoints of the opposite sides. The intersection of the bimedians is the centroid of the vertices of the quadrilateral. [13]

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 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. [31] 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 lengths 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 [21]

and

Note that the two opposite sides in these formulas are not the two that the bimedian connects.

In a convex quadrilateral, there is the following dual connection between the bimedians and the diagonals: [27]

Trigonometric identities

The four angles of a simple quadrilateral ABCD satisfy the following identities: [32]

and

Also, [33]

In the last two formulas, no angle is allowed to be a right angle, since tan 90° is not defined.

Inequalities

Area

If a convex quadrilateral has the consecutive sides a, b, c, d and the diagonals p, q, then its area K satisfies [34]

with equality only for a rectangle.
with equality only for a square.
with equality only if the diagonals are perpendicular and equal.
with equality only for a rectangle. [15]

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 [35]

Denoting the perimeter as L, we have [35] :p.114

with equality only in the case of a square.

The area of a convex quadrilateral also satisfies

for diagonal lengths p and q, with equality if and only if the diagonals are perpendicular.

Let a, b, c, d be the lengths of the sides of a convex quadrilateral ABCD with the area K and diagonals AC = p, BD = q. Then [36]

with equality only for a square.

Let a, b, c, d be the lengths of the sides of a convex quadrilateral ABCD with the area K, then the following inequality holds: [37]

with equality only for a square.

Diagonals and bimedians

A corollary to Euler's quadrilateral theorem 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.

In any convex quadrilateral the bimedians m, n and the diagonals p, q are related by the inequality

with equality holding if and only if the diagonals are equal. [38] :Prop.1 This follows directly from the quadrilateral identity

Sides

The sides a, b, c, and d of any quadrilateral satisfy [39] :p.228,#275

and [39] :p.234,#466

Maximum and minimum properties

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 [35] :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. [40]

Of all convex quadrilaterals with given diagonals, the orthodiagonal quadrilateral has the largest area. [35] :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. [41] :p.120

Remarkable points and lines in a convex quadrilateral

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. [42]

The "vertex centroid" is the intersection of the two bimedians. [43] 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 Ga, Gb, Gc, Gd be the centroids of triangles BCD, ACD, ABD, ABC respectively. Then the "area centroid" is the intersection of the lines GaGc and GbGd. [44]

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 Oa, Ob, Oc, Od be the circumcenters of triangles BCD, ACD, ABD, ABC respectively; and denote by Ha, Hb, Hc, Hd the orthocenters in the same triangles. Then the intersection of the lines OaOc and ObOd is called the quasicircumcenter, and the intersection of the lines HaHc and HbHd is called the quasiorthocenter of the convex quadrilateral. [44] 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. [44]

There can also be defined a quasinine-point centerE as the intersection of the lines EaEc and EbEd, where Ea, Eb, Ec, Ed are the nine-point centers of triangles BCD, ACD, ABD, ABC respectively. Then E is the midpoint of OH. [44]

Another remarkable line in a convex non-parallelogram quadrilateral is the Newton line, which connects the midpoints of the diagonals, the segment connecting these points being bisected by the vertex centroid. One more interesting line (in some sense dual to the Newton's one) is the line connecting the point of intersection of diagonals with the vertex centroid. The line is remarkable by the fact that it contains the (area) centroid. The vertex centroid divides the segment connecting the intersection of diagonals and the (area) centroid in the ratio 3:1. [45]

For any quadrilateral ABCD with points P and Q the intersections of AD and BC and AB and CD, respectively, the circles (PAB), (PCD), (QAD), and (QBC) pass through a common point M, called a Miquel point. [46]

For a convex quadrilateral ABCD in which E is the point of intersection of the diagonals and F is the point of intersection of the extensions of sides BC and AD, let ω be a circle through E and F which meets CB internally at M and DA internally at N. Let CA meet ω again at L and let DB meet ω again at K. Then there holds: the straight lines NK and ML intersect at point P that is located on the side AB; the straight lines NL and KM intersect at point Q that is located on the side CD. Points P and Q are called ”Pascal points” formed by circle ω on sides AB and CD. [47] [48] [49]

Other properties of convex quadrilaterals

Taxonomy

A taxonomy of quadrilaterals, using a Hasse diagram. Quadrilateral hierarchy svg.svg
A taxonomy of quadrilaterals, using a Hasse diagram.

A hierarchical taxonomy of quadrilaterals is illustrated by the figure to the right. Lower classes are special cases of higher classes they are connected to. Note that "trapezoid" here is referring to the North American definition (the British equivalent is a trapezium). Inclusive definitions are used throughout.

Skew quadrilaterals

The (red) side edges of tetragonal disphenoid represent a regular zig-zag skew quadrilateral Disphenoid tetrahedron.png
The (red) side edges of tetragonal disphenoid represent a regular zig-zag skew quadrilateral

A non-planar quadrilateral is called a skew quadrilateral. Formulas to compute its dihedral angles from the edge lengths and the angle between two adjacent edges were derived for work on the properties of molecules such as cyclobutane that contain a "puckered" ring of four atoms. [51] Historically the term gauche quadrilateral was also used to mean a skew quadrilateral. [52] A skew quadrilateral together with its diagonals form a (possibly non-regular) tetrahedron, and conversely every skew quadrilateral comes from a tetrahedron where a pair of opposite edges is removed.

See also

Related Research Articles

Triangle Shape with three sides

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted .

Rectangle Quadrilateral with four right angles

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as  ABCD.

Parallelogram

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

Bisection

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector.

Rhombus Quadrilateral in which all sides have the same length

In plane Euclidean geometry, a rhombus is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

In Euclidean geometry, Brahmagupta's formula is used to find the area of any cyclic quadrilateral given the lengths of the sides.

Cyclic quadrilateral Quadrilateral whose vertices can all fall on a single circle

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

Trapezoid Convex quadrilateral with at least one pair of parallel sides

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezium in English outside North America, but as a trapezoid in American and Canadian English. The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides. A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast to the special cases below.

In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be either convex or star. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.

Isosceles trapezoid Trapezoid symmetrical about an axis

In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of the same measure. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides are parallel, and the two other sides are of equal length. The diagonals are also of equal length. The base angles of an isosceles trapezoid are equal in measure.

Square Regular quadrilateral

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ABCD.

Bretschneiders formula

In geometry, Bretschneider's formula is the following expression for the area of a general quadrilateral:

Tangential quadrilateral

In Euclidean geometry, a tangential quadrilateral or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons.

Varignons theorem The midpoints of the sides of an arbitrary quadrilateral form a parallelogram

Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, whose proof was published posthumously in 1731.

Pentagon shape with five sides

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

Bicentric quadrilateral Type of shape

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral. It has also rarely been called a double circle quadrilateral and double scribed quadrilateral.

Orthodiagonal quadrilateral

In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

Ex-tangential quadrilateral

In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. It has also been called an exscriptible quadrilateral. The circle is called its excircle, its radius the exradius and its center the excenter. The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The ex-tangential quadrilateral is closely related to the tangential quadrilateral.

Equidiagonal quadrilateral

In Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified first according to whether they were equidiagonal and then into more specialized types.

Right kite Symmetrical quadrilateral

In Euclidean geometry, a right kite is a kite that can be inscribed in a circle. That is, it is a kite with a circumcircle. Thus the right kite is a convex quadrilateral and has two opposite right angles. If there are exactly two right angles, each must be between sides of different lengths. All right kites are bicentric quadrilaterals, since all kites have an incircle. One of the diagonals divides the right kite into two right triangles and is also a diameter of the circumcircle.

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