Quadrilateral

Template:Short description Template:About Template:Redirect Template:Infobox Polygon 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>.<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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<ref name=":0" />

<math>\angle A+\angle B+\angle C+\angle D=360^{\circ}.</math>

This is a special case of the n-gon interior angle sum formula: S = (n − 2) × 180° (here, n=4).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

All non-self-crossing quadrilaterals tile the plane, by repeated rotation around the midpoints of their edges.<ref>Template:Citation</ref>

Simple quadrilaterals

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

Convex quadrilateral

File:Euler diagram of quadrilateral types.svg

Euler diagram of some types of simple quadrilaterals. (UK) denotes British English and (US) denotes American English.

File: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.

• Irregular quadrilateral (British English) or trapezium (North American English): no sides are parallel. (In British English, this was once called a trapezoid. For more, see Template:Section link.)
• Trapezium (UK) or trapezoid (US): at least one pair of opposite sides are parallel. Trapezia (UK) and trapezoids (US) include parallelograms.
• 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:<ref name=":0" /> all four sides are of equal length (equilateral). An equivalent condition is that the diagonals perpendicularly bisect each other. Informally: "a pushed-over square" (but strictly including a square, too).
• Rhomboid: a parallelogram in which adjacent sides are of unequal lengths, and some angles are oblique (equiv., having no right angles). Informally: "a pushed-over oblong". Not all references agree; some define a rhomboid as a parallelogram that is not a rhombus.<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

• 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).<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

• 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.

• 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.<ref>Template:Cite journal</ref>
• A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square.<ref>Template:Cite journal</ref>
• A diametric quadrilateral is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle.<ref>Template:Cite journal</ref>
• A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices.<ref>Template:Cite book</ref>

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

File: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°.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

• Crossed trapezoid (US) or trapezium (Commonwealth):<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref> 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> They intersect at the "vertex centroid" of the quadrilateral (see Template:Section link below).

The four maltitudes of a convex quadrilateral are the perpendiculars to a side—through the midpoint of the opposite side.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Area of a convex quadrilateral

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

Trigonometric formulas

The area can be expressed in trigonometric terms as<ref name=":1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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

where the lengths of the diagonals are Template:Math and Template:Math and the angle between them is Template:Math.<ref>Harries, J. "Area of a quadrilateral," Mathematical Gazette 86, July 2002, 310–311.</ref> In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to <math>K=\tfrac{pq}{2}</math> since Template:Math is Template:Math.

The area can be also expressed in terms of bimedians as<ref name=Josefsson4/>

<math>K = mn \sin \varphi,</math>

where the lengths of the bimedians are Template:Math and Template:Math and the angle between them is Template:Math.

Bretschneider's formula<ref>R. A. Johnson, Advanced Euclidean Geometry, 2007, Dover Publ., p. 82.</ref><ref name=":1" /> expresses the area in terms of the sides and two opposite angles:

<math>\begin{align}

K &= \sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{2} abcd \; [ 1 + \cos (A + C) ]} \\ &= \sqrt{(s-a)(s-b)(s-c)(s-d) - abcd\, \cos^2 \tfrac12(A + C) } \end{align}</math>

where the sides in sequence are Template:Math, Template:Math, Template:Math, Template:Math, where Template:Math is the semiperimeter, and Template:Math and Template:Math are two (in fact, any two) opposite angles. This reduces to Brahmagupta's formula for the area of a cyclic quadrilateral—when Template:Math.

Another area formula in terms of the sides and angles, with angle Template:Math being between sides Template:Math and Template:Math, and Template:Math being between sides Template:Math and Template:Math, is

<math>K = \tfrac12 ad \sin{A} + \tfrac12 bc \sin{C}.</math>

In the case of a cyclic quadrilateral, the latter formula becomes <math>K = \tfrac12(ad+bc)\sin{A}.</math>

In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to <math>K=ab \cdot \sin{A}.</math>

Alternatively, we can write the area in terms of the sides and the intersection angle Template:Math of the diagonals, as long as Template:Math is not Template:Math:<ref name=Mitchell>Mitchell, Douglas W., "The area of a quadrilateral," Mathematical Gazette 93, July 2009, 306–309.</ref>

<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 Template:Math, Template:Math, Template:Math, Template:Math is<ref name=Josefsson4>Template:Citation.</ref>

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

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

The last trigonometric area formula including the sides Template:Math, Template:Math, Template:Math, Template:Math and the angle Template:Math (between Template:Math and Template:Math) is:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<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 Template:Math), by just changing the first sign Template:Math to Template:Math.

Non-trigonometric formulas

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

<math>K = \sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{4}(ac+bd+pq)(ac+bd-pq)},</math> <ref>J. L. Coolidge, "A historically interesting formula for the area of a quadrilateral", American Mathematical Monthly, 46 (1939) 345–347.</ref>

<math>K = \tfrac14 \sqrt{4p^2q^2 - \left( a^2 + c^2 - b^2 - d^2 \right)^2}.</math> <ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

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

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

<math>K=\tfrac12 \sqrt{(m+n+p)(m+n-p)(m+n+q)(m+n-q)},</math> <ref>Archibald, R. C., "The Area of a Quadrilateral", American Mathematical Monthly, 29 (1922) pp. 29–36.</ref>

<math>K=\tfrac12 \sqrt{p^2q^2-(m^2-n^2)^2}.</math> <ref name=Josefsson3>Template:Citation.</ref>Template:Rp

In fact, any three of the four values Template:Math, Template:Math, Template:Math, and Template:Math 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><ref name=Altshiller-Court/>Template:Rp The corresponding expressions are:<ref name=Josefsson6>Josefsson, Martin (2016) ‘100.31 Heron-like formulas for quadrilaterals’, The Mathematical Gazette, 100 (549), pp. 505–508.</ref>

<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<ref name=Josefsson6/>

<math>K=\tfrac14 \sqrt{[(p+q)^2-4m^2]\cdot[4m^2-(p-q)^2]},</math>

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

Vector formulas

The area of a quadrilateral Template:Math can be calculated using vectors. Let vectors Template:Math and Template:Math form the diagonals from Template:Math to Template:Math and from Template:Math to Template:Math. The area of the quadrilateral is then

<math>K = \tfrac12 |\mathbf{AC}\times\mathbf{BD}|,</math>

which is half the magnitude of the cross product of vectors Template:Math and Template:Math. In two-dimensional Euclidean space, expressing vector Template:Math as a free vector in Cartesian space equal to Template:Math and Template:Math as Template:Math, this can be rewritten as:

<math>K = \tfrac12 |x_1 y_2 - x_2 y_1|.</math>

Diagonals

Properties of the diagonals in 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The list applies to the most general cases, and excludes named subsets.

Quadrilateral	Bisecting diagonals	Perpendicular diagonals	Equal diagonals
Trapezoid	Template:No	See note 1	Template:No
Isosceles trapezoid	Template:No	See note 1	Template:Yes
Parallelogram	Template:Yes	Template:No	Template:No
Kite	See note 2	Template:Yes	See note 2
Rectangle	Template:Yes	Template:No	Template:Yes
Rhombus	Template:Yes	Template:Yes	Template:No
Square	Template:Yes	Template:Yes	Template:Yes

• 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<ref>Rashid, M. A. & Ajibade, A. O., "Two conditions for a quadrilateral to be cyclic expressed in terms of the lengths of its sides", Int. J. Math. Educ. Sci. Technol., vol. 34 (2003) no. 5, pp. 739–799.</ref>

<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 Template:Mvar is the distance between the midpoints of the diagonals.<ref name=Altshiller-Court/>Template:Rp 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<ref>Andreescu, Titu & Andrica, Dorian, Complex Numbers from A to...Z, Birkhäuser, 2006, pp. 207–209.</ref>

<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 Template:Math, it reduces to Template:Math. Since Template:Math, it also gives a proof of Ptolemy's inequality.

Other metric relations

If Template:Mvar and Template:Mvar are the feet of the normals from Template:Mvar and Template:Mvar to the diagonal Template:Math in a convex quadrilateral ABCD with sides Template:Math, Template:Math, Template:Math, Template:Math, then<ref name=Josefsson/>Template:Rp

<math>XY=\frac{|a^2+c^2-b^2-d^2|}{2p}.</math>

In a convex quadrilateral ABCD with sides Template:Math, Template:Math, Template:Math, Template:Math, and where the diagonals intersect at Template:Mvar,

<math> efgh(a+c+b+d)(a+c-b-d) = (agh+cef+beh+dfg)(agh+cef-beh-dfg)</math>

where Template:Math, Template:Math, Template:Math, and Template:Math.<ref>Template:Citation.</ref>

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 Template:Math and the four side lengths Template:Math of a quadrilateral are related<ref name=":1" /> by the Cayley-Menger determinant, as follows:

<math> \det \begin{bmatrix}

  0 & a^2 & p^2 & d^2 & 1 \\
a^2 &   0 & b^2 & q^2 & 1 \\
p^2 & b^2 &   0 & c^2 & 1 \\
d^2 & q^2 & c^2 &   0 & 1 \\
  1 &   1 &   1 & 1   & 0

\end{bmatrix} = 0. </math>

Angle bisectors

The internal angle bisectors of a convex quadrilateral either form a cyclic quadrilateral<ref name=Altshiller-Court/>Template:Rp (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 Template:Mvar and Template:Mvar meet on diagonal Template:Mvar, then the angle bisectors of Template:Mvar and Template:Mvar meet on diagonal Template:Mvar.<ref>Leversha, Gerry, "A property of the diagonals of a cyclic quadrilateral", Mathematical Gazette 93, March 2009, 116–118.</ref>

Bimedians

File: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.<ref name=":1" />

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:

• Each pair of opposite sides of the Varignon parallelogram are parallel to a diagonal in the original quadrilateral.
• A side of the Varignon parallelogram is half as long as the diagonal in the original quadrilateral it is parallel to.
• The area of the Varignon parallelogram equals half the area of the original quadrilateral. This is true in convex, concave and crossed quadrilaterals provided the area of the latter is defined to be the difference of the areas of the two triangles it is composed of.<ref>H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, MAA, 1967, pp. 52–53.</ref>
• 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.<ref name=Altshiller-Court>Altshiller-Court, Nathan, College Geometry, Dover Publ., 2007.</ref>Template:Rp

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

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

where Template:Mvar and Template:Mvar are the length of the diagonals.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The length of the bimedian that connects the midpoints of the sides Template:Mvar and Template:Mvar is

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

Hence<ref name=Altshiller-Court/>Template:Rp

<math>\displaystyle p^2+q^2=2(m^2+n^2).</math>

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 Template:Mvar between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence<ref name=Josefsson3/>

<math>m=\tfrac{1}{2}\sqrt{2(b^2+d^2)-4x^2}</math>

and

<math>n=\tfrac{1}{2}\sqrt{2(a^2+c^2)-4x^2}.</math>

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:<ref name=Josefsson>Template:Citation.</ref>

• 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:<ref>C. V. Durell & A. Robson, Advanced Trigonometry, Dover, 2003, p. 267.</ref>

<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,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<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 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<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<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<ref>O. Bottema, Geometric Inequalities, Wolters–Noordhoff Publishing, The Netherlands, 1969, pp. 129, 132.</ref>

<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.<ref name=Josefsson4/>

From Bretschneider's formula it directly follows that the area of a quadrilateral satisfies

<math>K \le \sqrt{(s-a)(s-b)(s-c)(s-d)}</math>

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).

Also,

<math>K \leq \sqrt{abcd},</math>

with equality for a bicentric quadrilateral or a rectangle.

The area of any quadrilateral also satisfies the inequality<ref name=Alsina>Template:Citation.</ref>

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

Denoting the perimeter as L, we have<ref name=Alsina/>Template:Rp

<math>K\le \tfrac{1}{16}L^2,</math>

with equality only in the case of a square.

The area of a convex quadrilateral also satisfies

<math>K \le \tfrac{1}{2}pq</math>

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<ref>Dao Thanh Oai, Leonard Giugiuc, Problem 12033, American Mathematical Monthly, March 2018, p. 277</ref>

<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:<ref>Template:Cite journal</ref>

<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.<ref name=Altshiller-Court/>Template:Rp 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

<math>pq \leq m^2+n^2,</math>

with equality holding if and only if the diagonals are equal.<ref name=J2014>Template:Cite journal</ref>Template:Rp 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<ref name=Crux>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>Template:Rp

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

and<ref name=Crux/>Template:Rp

<math>a^4+b^4+c^4 \geq \tfrac1{27} d^4.</math>

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<ref name=Alsina/>Template:Rp

<math>K\le \tfrac{1}{16}L^2</math>

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.<ref name=Peter/>

Of all convex quadrilaterals with given diagonals, the orthodiagonal quadrilateral has the largest area.<ref name=Alsina/>Template:Rp This is a direct consequence of the fact that the area of a convex quadrilateral satisfies

<math>K=\tfrac{1}{2}pq\sin{\theta}\le \tfrac{1}{2}pq,</math>

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

<math>AP+BP+CP+DP\ge AC+BD.</math>

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.<ref name=autogenerated1>Template:Cite book</ref>Template:Rp

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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

The "vertex centroid" is the intersection of the two bimedians.<ref>Honsberger, Ross, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Math. Assoc. Amer., 1995, pp. 35–41.</ref> 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_aG_c and G_bG_d.<ref name=Myakishev>Template:Citation.</ref>

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_aO_c and O_bO_d is called the quasicircumcenter, and the intersection of the lines H_aH_c and H_bH_d is called the quasiorthocenter of the convex quadrilateral.<ref name=Myakishev/> 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.<ref name=Myakishev/>

There can also be defined a quasinine-point center E as the intersection of the lines E_aE_c and E_bE_d, where E_a, E_b, E_c, E_d are the nine-point centers of triangles BCD, ACD, ABD, ABC respectively. Then E is the midpoint of OH.<ref name=Myakishev/>

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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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.<ref>Template:Cite book</ref>

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. <ref name=Fraivert>Template:Citation.</ref> <ref name=Fraivert2>Template:Citation.</ref> <ref name=Fraivert3>Template:Citation.</ref>

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.<ref name=Peter>Peter, Thomas, "Maximizing the Area of a Quadrilateral", The College Mathematics Journal, Vol. 34, No. 4 (September 2003), pp. 315–316.</ref>
• 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.<ref>Template:Cite journal</ref>
• The angle <math>\theta</math> at the intersection of the diagonals satisfies <math display=block>\cos \theta = \frac{a^2+c^2-b^2-d^2}{2pq},</math> where <math>p, q</math> are the diagonals of the quadrilateral.<ref>Template:Cite book</ref>

Taxonomy

File: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

File: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.<ref>Template:Cite journal</ref> Historically the term gauche quadrilateral was also used to mean a skew quadrilateral.<ref>Template:Cite journal</ref> 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
• Template:Section link
• Quadrangle (geography)
• Homography - Any quadrilateral can be transformed into another quadrilateral by a projective transformation (homography)

External links

• Template:Springer
• 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 Template:Webarchive at Dynamic Math Learning Homepage Template:Webarchive
• The role and function of a hierarchical classification of quadrilaterals by Michael de Villiers

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