In geometry, a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. pentagon). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> is sometimes denoted as <math>\square ABCD</math>.

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, that is

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 quadrilateral

thumb|upright=1.35|[[Euler diagram of some types of simple quadrilaterals. (UK) denotes British English and (US) denotes American English.]]

upright=1.2|thumb|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.

  • Irregular quadrilateral: no sides are parallel.
  • Trapezium (UK) or trapezoid (US): at least one pair of opposite sides are parallel. Trapezia (UK) and trapezoids (US) include parallelograms.

<!--Please do NOT define an isosceles trapezoid as having legs equal. Doing so would make all parallelograms isosceles trapezoids, which we know is wrong.-->

  • Isosceles trapezium (UK) or isosceles trapezoid (US): one pair of opposite sides are parallel and the base angles are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, or a trapezoid with diagonals of equal length.
  • Parallelogram: 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 include rhombi (including those rectangles called squares) and rhomboids (including those rectangles called oblongs). In other words, parallelograms include all rhombi and all rhomboids, and thus also include all rectangles.
  • Rhombus, rhomb:
  • Rectangle: all four angles are right angles (equiangular). An equivalent condition is that the diagonals bisect each other, and are equal in length. Rectangles include squares and oblongs. Informally: "a box or oblong" (including a square).
  • Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle (i.e., four equal sides and four equal angles).
  • Oblong: longer than wide, or wider than long (i.e., a rectangle that is not a square).
  • Kite: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into congruent triangles, and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular. Kites include rhombi.

File:Quadrilaterals.svg

  • Tangential quadrilateral: the four sides are tangents to an inscribed circle. A convex quadrilateral is tangential if and only if opposite sides have equal sums.
  • Tangential trapezoid: a trapezoid where the four sides are tangents to an inscribed circle.
  • Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.
  • Right kite: a kite with two opposite right angles. It is a type of cyclic quadrilateral.
  • Harmonic quadrilateral: a cyclic quadrilateral such that the products of the lengths of the opposing sides are equal.
  • Bicentric quadrilateral: it is both tangential and cyclic.
  • Orthodiagonal quadrilateral: the diagonals cross at right angles.
  • Equidiagonal quadrilateral: the diagonals are of equal length.
  • Bisect-diagonal quadrilateral: one diagonal bisects the other into equal lengths. Every dart and kite is bisect-diagonal. When both diagonals bisect another, it's a parallelogram.
  • Ex-tangential quadrilateral: the four extensions of the sides are tangent to an excircle.
  • An equilic quadrilateral has two opposite equal sides that when extended, meet at 60°.
  • A Watt quadrilateral is a quadrilateral with a pair of opposite sides of equal length.
  • A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square.
  • A diametric quadrilateral is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle.
  • A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices.

Concave quadrilaterals

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

  • A dart (or arrowhead) is a concave quadrilateral with bilateral symmetry like a kite, but where one interior angle is reflex. See Kite.

Complex quadrilaterals

thumb|upright=0.8|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°.

  • Crossed trapezoid (US) or trapezium (Commonwealth): a crossed quadrilateral in which one pair of nonadjacent sides is parallel (like a trapezoid).
  • Antiparallelogram: a crossed quadrilateral in which each pair of nonadjacent sides have equal lengths (like a parallelogram).
  • Crossed rectangle: an antiparallelogram whose sides are two opposite sides and the two diagonals of a rectangle, hence having one pair of parallel opposite sides.
  • Crossed square: a special case of a crossed rectangle where two of the sides intersect at right angles.

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. They intersect at the "vertex centroid" of the quadrilateral (see below).

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

Area of a convex quadrilateral

There are various general formulas for the area of a convex quadrilateral ABCD with sides .

Trigonometric formulas

The area can be expressed in trigonometric terms as

:<math>K = \tfrac12 pq \sin \theta,</math>

where the lengths of the diagonals are and and the angle between them is . In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to <math>K=\tfrac{pq}{2}</math> since is .

The area can be also expressed in terms of bimedians as

:<math>K = \tfrac14 \left|\tan \theta\right| \cdot \left| a^2 + c^2 - b^2 - d^2 \right|.</math>

In the case of a parallelogram, the latter formula becomes <math>K = \tfrac12 \left|\tan \theta\right| \cdot \left| a^2 - b^2 \right|.</math>

Another area formula including the sides , , , is

:<math>K=\tfrac12 \sqrt{\bigl((a^2+c^2)-2x^2\bigr)\bigl((b^2+d^2)-2x^2\bigr)} \sin{\varphi}</math>

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

The last trigonometric area formula including the sides , , , and the angle (between and ) is:

:<math>K=\tfrac12 ab \sin{\alpha}+\tfrac14 \sqrt{4c^2d^2-(c^2+d^2-a^2-b^2+2ab \cos{\alpha})^2} ,</math>

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

Non-trigonometric formulas

The following two formulas express the area in terms of the sides , , and , the semiperimeter , and the diagonals , :

:<math>K = \sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{4}(ac+bd+pq)(ac+bd-pq)},</math>

:<math>K = \tfrac14 \sqrt{4p^2q^2 - \left( a^2 + c^2 - b^2 - d^2 \right)^2}.</math>

The first reduces to Brahmagupta's formula in the cyclic quadrilateral case, since then .

The area can also be expressed in terms of the bimedians , and the diagonals , :

:<math>K=\tfrac12 \sqrt{(m+n+p)(m+n-p)(m+n+q)(m+n-q)},</math>

:<math>K=\tfrac12 \sqrt{p^2q^2-(m^2-n^2)^2}.</math>

In fact, any three of the four values , , , and suffice for determination of the area, since in any quadrilateral the four values are related by <math>p^2+q^2=2(m^2+n^2).</math>

:<math>K=\tfrac12 \sqrt{[(m+n)^2-p^2]\cdot[p^2-(m-n)^2]},</math>

if the lengths of two bimedians and one diagonal are given, and The list applies to the most general cases, and excludes named subsets.

{| class="wikitable"

|-

! scope="col" | Quadrilateral

! scope="col" | Bisecting diagonals

! scope="col" | Perpendicular diagonals

! scope="col" | Equal diagonals

|-

! scope="row" | Trapezoid

| || See note 1 ||

|-

! scope="row" | Isosceles trapezoid

| || See note 1 ||

|-<!--

! scope="row" | Right trapezoid

|| See note 3 || See note 1 ||

|--->

! scope="row" | Parallelogram

| || ||

|-

! scope="row" | Kite

| See note 2 || || See note 2

|-

! scope="row" | Rectangle

| || ||

|-

! scope="row" | Rhombus

| || ||

|-

! scope="row" | Square

| || ||

|}

  • 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

:<math>p=\sqrt{a^2+b^2-2ab\cos{B=\sqrt{c^2+d^2-2cd\cos{D</math>

and

:<math>q=\sqrt{a^2+d^2-2ad\cos{A=\sqrt{b^2+c^2-2bc\cos{C.</math>

Other, more symmetric formulas for the lengths of the diagonals, are

:<math>p=\sqrt{\frac{(ac+bd)(ad+bc)-2abcd(\cos{B}+\cos{D})}{ab+cd</math>

and

:<math>q=\sqrt{\frac{(ab+cd)(ac+bd)-2abcd(\cos{A}+\cos{C})}{ad+bc.</math>

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

:<math> a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4x^2 </math>

where is the distance between the midpoints of the diagonals.

:<math> p^2q^2=a^2c^2+b^2d^2-2abcd\cos{(A+C)}.</math>

This relation can be considered to be a law of cosines for a quadrilateral. In a cyclic quadrilateral, where , it reduces to . Since , it also gives a proof of Ptolemy's inequality.

Other metric relations

If and are the feet of the normals from and to the diagonal in a convex quadrilateral ABCD with sides , , , , then

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 and the four side lengths of a quadrilateral are related

Bimedians

[[File:Varignon theorem convex.png|upright=1.2|thumb|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.

  • The perimeter of the Varignon parallelogram equals the sum of the diagonals of the original quadrilateral.
  • 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.

In a convex quadrilateral with sides , , and , the length of the bimedian that connects the midpoints of the sides and is

:<math>m=\tfrac{1}{2}\sqrt{-a^2+b^2-c^2+d^2+p^2+q^2}</math>

where and are the length of the diagonals. The length of the bimedian that connects the midpoints of the sides and is

:<math>n=\tfrac{1}{2}\sqrt{a^2-b^2+c^2-d^2+p^2+q^2}.</math>

Hence

  • The two bimedians have equal length if and only if the two diagonals are perpendicular.
  • The two bimedians are perpendicular if and only if the two diagonals have equal length.

Trigonometric identities

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

:<math>\sin A + \sin B + \sin C + \sin D = 4\sin\tfrac12(A+B)\, \sin\tfrac12(A+C)\, \sin\tfrac12(A+D)</math>

and

:<math>

\frac{\tan A\,\tan{B} - \tan C\,\tan D}{\tan A\,\tan C - \tan B\,\tan D}

= \frac{\tan(A+C)}{\tan(A+B)}.

</math>

Also,

:<math>

\frac{\tan A + \tan B + \tan C + \tan D}{\cot A + \cot B + \cot C + \cot D}

= \tan{A}\tan{B}\tan{C}\tan{D}.

</math>

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

Let <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> be the sides of a convex quadrilateral, <math>s</math> is the semiperimeter,

and <math>A</math> and <math>C</math> are opposite angles, then

:<math>ad\sin^2\tfrac12 A + bc\cos^2\tfrac12C = (s-a)(s-d)</math>

and

:<math>bc\sin^2\tfrac12 C + ad\cos^2\tfrac12 A = (s-b)(s-c)</math>.

We can use these identities to derive the Bretschneider's Formula.

Inequalities

Area

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

:<math>K\le \tfrac{1}{4}(a+c)(b+d)</math> with equality only for a rectangle.

:<math>K\le \tfrac{1}{4}(a^2+b^2+c^2+d^2)</math> with equality only for a square.

:<math>K\le \tfrac{1}{4}(p^2+q^2)</math> with equality only if the diagonals are perpendicular and equal.

:<math>K\le \tfrac{1}{2}\sqrt{(a^2+c^2)(b^2+d^2)}</math> with equality only for a rectangle.

:<math>\displaystyle K\le \tfrac{1}{2}\sqrt[3]{(ab+cd)(ac+bd)(ad+bc)}.</math>

Denoting the perimeter as L, we have

:<math> K \leq \tfrac18(a^2+b^2+c^2+d^2+p^2+q^2+pq-ac-bd) </math> 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:

:<math> K \leq \frac{1}{3+\sqrt{3(ab+ac+ad+bc+bd+cd) - \frac{1}{2(1+\sqrt{3})^2}(a^2+b^2+c^2+d^2) </math> with equality only for a square.

Diagonals and bimedians

A corollary to Euler's quadrilateral theorem is the inequality

:<math> a^2 + b^2 + c^2 + d^2 \ge p^2 + q^2 </math>

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

:<math> pq \le ac + bd </math>

where there is equality if and only if the quadrilateral is cyclic. This follows directly from the quadrilateral identity <math>m^2+n^2=\tfrac{1}{2}(p^2+q^2).</math>

Sides

The sides a, b, c, and d of any quadrilateral satisfy

:<math>a^2+b^2+c^2 > \tfrac13 d^2</math>

and

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.

The "vertex centroid" is the intersection of the two bimedians. 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<sub>a</sub>, G<sub>b</sub>, G<sub>c</sub>, G<sub>d</sub> be the centroids of triangles BCD, ACD, ABD, ABC respectively. Then the "area centroid" is the intersection of the lines G<sub>a</sub>G<sub>c</sub> and G<sub>b</sub>G<sub>d</sub>.

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<sub>a</sub>, O<sub>b</sub>, O<sub>c</sub>, O<sub>d</sub> be the circumcenters of triangles BCD, ACD, ABD, ABC respectively; and denote by H<sub>a</sub>, H<sub>b</sub>, H<sub>c</sub>, H<sub>d</sub> the orthocenters in the same triangles. Then the intersection of the lines O<sub>a</sub>O<sub>c</sub> and O<sub>b</sub>O<sub>d</sub> is called the quasicircumcenter, and the intersection of the lines H<sub>a</sub>H<sub>c</sub> and H<sub>b</sub>H<sub>d</sub> is called the quasiorthocenter of the convex quadrilateral.

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.

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, applying Pascal's theorem to the hexagons EKNFML and EKMFNL inscribed in ω, 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.

Other properties of convex quadrilaterals

  • If exterior squares are drawn on all sides of a quadrilateral then the segments connecting the centers of opposite squares are (a) equal in length, and (b) perpendicular. Thus these centers are the vertices of an orthodiagonal quadrilateral. This is called Van Aubel's theorem.
  • For any simple quadrilateral with given edge lengths, there is a cyclic quadrilateral with the same edge lengths.
  • The four smaller triangles formed by the diagonals and sides of a convex quadrilateral have the property that the product of the areas of two opposite triangles equals the product of the areas of the other two triangles.
  • Let vectors and form the diagonals from A to C and from B to D. The angle <math>\theta</math> at the intersection of the diagonals satisfies <math display="block">\cos \theta = \frac{b^2+d^2-a^2-c^2}{2pq},</math> where <math>\theta</math> is the angle between and , and <math>p, q</math> are the diagonals of the quadrilateral.

Taxonomy

upright=1.2|thumb|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

thumb|The (red) side edges of [[tetragonal disphenoid represent a regular zig-zag skew quadrilateral.]]

A generalization of ordinary quadrilaterals to non-planar figures is called a skew quadrilateral, a four-sided skew polygon. 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. Historically the term gauche quadrilateral was also used to mean a skew quadrilateral. 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

  • Complete quadrangle
  • Perpendicular bisector construction of a quadrilateral
  • Saccheri quadrilateral
  • Quadrangle (geography)
  • Homography - Any quadrilateral can be transformed into another quadrilateral by a projective transformation (homography)

References

  • Quadrilaterals Formed by Perpendicular Bisectors, Projective Collinearity and Interactive Classification of Quadrilaterals from cut-the-knot
  • Definitions and examples of quadrilaterals and Definition and properties of tetragons from Mathopenref
  • A (dynamic) Hierarchical Quadrilateral Tree at Dynamic Geometry Sketches
  • An extended classification of quadrilaterals at Dynamic Math Learning Homepage
  • The role and function of a hierarchical classification of quadrilaterals by Michael de Villiers