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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
a quadrilateral is a four-sided
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
, 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 In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be sim ...
). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices A, B, C and D is sometimes denoted as \square ABCD. Quadrilaterals are either
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
(not self-intersecting), or complex (self-intersecting, or crossed). Simple quadrilaterals are either convex or
concave Concave or concavity may refer to: Science and technology * Concave lens * Concave mirror Mathematics * Concave function, the negative of a convex function * Concave polygon, a polygon which is not convex * Concave set In geometry, a subset o ...
. The interior angles of a simple (and planar) quadrilateral ''ABCD'' add up to 360 degrees of arc, that is :\angle A+\angle B+\angle C+\angle D=360^. This is a special case of the ''n''-gon interior angle sum formula: ''S'' = (''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 quadrilateral

In a convex quadrilateral all interior angles are less than 180°, and the two diagonals both lie inside the quadrilateral. *Irregular quadrilateral (
British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, "English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the English language in England, or, more broadl ...
) or trapezium (
North American English North American English (NAmE, NAE) is the most generalized variety of the English language as spoken in the United States and Canada. Because of their related histories and cultures, plus the similarities between the pronunciations (accents), ...
): no sides are parallel. (In British English, this was once called a ''trapezoid''. For more, see ) * Trapezium (UK) or
trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...
(US): at least one pair of opposite sides are
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
. Trapezia (UK) and trapezoids (US) include parallelograms. * Isosceles trapezium (UK) or
isosceles trapezoid In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) 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 defin ...
(US): one pair of opposite sides are parallel and the base
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s 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 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 eq ...
: 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: 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. *
Rectangle 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 (360°/4 = 90°); or a parallelogram contain ...
: 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. *
Tangential quadrilateral In Euclidean geometry, a tangential quadrilateral (sometimes just tangent 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 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
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
s to an
inscribed circle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incen ...
. * Cyclic quadrilateral: the four vertices lie on a
circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°. *
Right kite In Euclidean geometry, a right kite is a kite (a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other) that can be inscribed in a circle.Michael de Villiers, ''Some Adventures in Eu ...
: a kite with two opposite right angles. It is a type of cyclic quadrilateral. * Harmonic quadrilateral: the products of the lengths of the opposing sides are equal. It is a type of cyclic quadrilateral. * Bicentric quadrilateral: it is both tangential and cyclic. * Orthodiagonal quadrilateral: the diagonals cross at
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...
s. * Equidiagonal quadrilateral: the diagonals are of equal length. * 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 Concave or concavity may refer to: Science and technology * Concave lens * Concave mirror Mathematics * Concave function, the negative of a convex function * Concave polygon, a polygon which is not convex * Concave set In geometry, a subset o ...
quadrilateral with bilateral symmetry like a kite, but where one interior angle is reflex. See Kite.


Complex quadrilaterals

A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed quadrilateral,
butterfly Butterflies are insects in the macrolepidopteran clade Rhopalocera from the order Lepidoptera, which also includes moths. Adult butterflies have large, often brightly coloured wings, and conspicuous, fluttering flight. The group compris ...
quadrilateral or bow-tie quadrilateral. In a crossed quadrilateral, the four "interior" angles on either side of the crossing (two
acute Acute may refer to: Science and technology * Acute angle ** Acute triangle ** Acute, a leaf shape in the glossary of leaf morphology * Acute (medicine), a disease that it is of short duration and of recent onset. ** Acute toxicity, the adverse ef ...
and two
reflex In biology, a reflex, or reflex action, is an involuntary, unplanned sequence or action and nearly instantaneous response to a stimulus. Reflexes are found with varying levels of complexity in organisms with a nervous system. A reflex occurs ...
, 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 A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...
) *
Antiparallelogram In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel. Instead, sides in the ...
: a crossed quadrilateral in which each pair of nonadjacent sides have equal lengths (like a
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 eq ...
) *
Crossed rectangle 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 (360°/4 = 90°); or a parallelogram containin ...
: an antiparallelogram whose sides are two opposite sides and the two diagonals of a
rectangle 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 (360°/4 = 90°); or a parallelogram contain ...
, 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 segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
s 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 Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
of a convex quadrilateral ''ABCD'' with sides .


Trigonometric formulas

The area can be expressed in trigonometric terms as :K = \frac \sin \theta, 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 K=\tfrac since is . The area can be also expressed in terms of bimedians as :K = mn \sin \varphi, where the lengths of the bimedians are and and the angle between them is . Bretschneider's formula expresses the area in terms of the sides and two opposite angles: :\begin K &= \sqrt \\ &= \sqrt \end where the sides in sequence are , , , , where is the semiperimeter, and and are two (in fact, any two) opposite angles. This reduces to Brahmagupta's formula for the area of a cyclic quadrilateral—when . Another area formula in terms of the sides and angles, with angle being between sides and , and being between sides and , is :K = \frac \sin + \frac \sin. In the case of a cyclic quadrilateral, the latter formula becomes K = \frac\sin. In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to K=ab \cdot \sin. Alternatively, we can write the area in terms of the sides and the intersection angle of the diagonals, as long is not :Mitchell, Douglas W., "The area of a quadrilateral," ''Mathematical Gazette'' 93, July 2009, 306–309. :K = \frac \cdot \left, a^2 + c^2 - b^2 - d^2 \. In the case of a parallelogram, the latter formula becomes K = \frac\cdot \left, a^2 - b^2 \. Another area formula including the sides , , , is. :K=\frac\sin 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: :K=\frac\sin+\frac , 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 In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate ...
, and the diagonals , : :K = \sqrt, :K = \frac. 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 , : :K=\frac, :K=\frac. . 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 p^2+q^2=2(m^2+n^2). The corresponding expressions are:Josefsson, Martin (2016) ‘100.31 Heron-like formulas for quadrilaterals’, ''The Mathematical Gazette'', 100 (549), pp. 505–508. :K=\frac, if the lengths of two bimedians and one diagonal are given, and :K=\frac, if the lengths of two diagonals and one bimedian are given.


Vector formulas

The area of a quadrilateral can be calculated using vectors. Let vectors and form the diagonals from to and from to . The area of the quadrilateral is then :K = \frac, which is half the magnitude of the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of vectors and . In two-dimensional Euclidean space, expressing vector as a free vector in Cartesian space equal to and as , this can be rewritten as: :K = \frac.


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 In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
, and if their diagonals have equal length. The list applies to the most general cases, and excludes named subsets. ''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 :p=\sqrt=\sqrt and :q=\sqrt=\sqrt. Other, more symmetric formulas for the lengths of the diagonals, are :p=\sqrt and :q=\sqrt.


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 : a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4x^2 where ''x'' is the distance between the midpoints of the diagonals. This is sometimes known as Euler's quadrilateral theorem and is a generalization of the parallelogram law. The German mathematician
Carl Anton Bretschneider Carl Anton Bretschneider (27 May 1808 – 6 November 1878) was a mathematician from Gotha, Germany. Bretschneider worked in geometry, number theory, and history of geometry. He also worked on logarithmic integrals and mathematical tables. He was ...
derived in 1842 the following generalization of
Ptolemy's theorem In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematici ...
, regarding the product of the diagonals in a convex quadrilateral : p^2q^2=a^2c^2+b^2d^2-2abcd\cos. 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 :XY=\frac. In a convex quadrilateral ''ABCD'' with sides ''a'' = ''AB'', ''b'' = ''BC'', ''c'' = ''CD'', ''d'' = ''DA'', and where the diagonals intersect at ''E'', : efgh(a+c+b+d)(a+c-b-d) = (agh+cef+beh+dfg)(agh+cef-beh-dfg) where ''e'' = ''AE'', ''f'' = ''BE'', ''g'' = ''CE'', and ''h'' = ''DE''. 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 by the Cayley-Menger
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
, as follows: : \det \begin 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 = 0.


Angle bisectors

The internal angle bisectors of a convex quadrilateral either form a cyclic quadrilateral (that is, the four intersection points of adjacent angle bisectors are
concyclic In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. All concyclic points are at the same distance from the center of the circle. Three points in the plane that do not all fall on a straight line ...
) or they are concurrent. In the latter case the quadrilateral is a
tangential quadrilateral In Euclidean geometry, a tangential quadrilateral (sometimes just tangent 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 ...
. 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''.


Bimedians

The bimedians of a quadrilateral are the line segments connecting the midpoints of the opposite sides. The intersection of the bimedians is the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
of the vertices of the quadrilateral. The midpoints of the sides of any quadrilateral (convex, concave or crossed) are the vertices of a
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 eq ...
called the
Varignon 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 ...
. 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. *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.Altshiller-Court, Nathan, ''College Geometry'', Dover Publ., 2007. 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 :m=\tfrac\sqrt where ''p'' and ''q'' are the length of the diagonals. The length of the bimedian that connects the midpoints of the sides ''b'' and ''d'' is :n=\tfrac\sqrt. Hence :\displaystyle p^2+q^2=2(m^2+n^2). This is also a
corollary In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
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 :m=\tfrac\sqrt and :n=\tfrac\sqrt. 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:. * The two bimedians have equal length
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
the two diagonals are
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
. * 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: :\sin+\sin+\sin+\sin=4\sin\sin\sin and :\frac=\frac. Also, :\frac=\tan\tan\tan\tan. In the last two formulas, no angle is allowed to be a
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...
, since tan 90° is not defined. Let a, b, c, d be the sides of a convex quadrilateral, s is the semiperimeter, and A and C are opposite angles, then :ad\sin^2+bc\cos^2=(s-a)(s-d) and :bc\sin^2+ad\cos^2=(s-b)(s-c). 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 :K\le \tfrac(a+c)(b+d) with equality only for a
rectangle 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 (360°/4 = 90°); or a parallelogram contain ...
. :K\le \tfrac(a^2+b^2+c^2+d^2) with equality only for a square. :K\le \tfrac(p^2+q^2) with equality only if the diagonals are perpendicular and equal. :K\le \tfrac\sqrt with equality only for a rectangle. From Bretschneider's formula it directly follows that the area of a quadrilateral satisfies :K \le \sqrt with equality
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
the quadrilateral is
cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...
or degenerate such that one side is equal to the sum of the other three (it has collapsed into a
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
, so the area is zero). The area of any quadrilateral also satisfies the inequality. :\displaystyle K\le \tfrac\sqrt Denoting the perimeter as ''L'', we have :K\le \tfracL^2, with equality only in the case of a square. The area of a convex quadrilateral also satisfies :K \le \tfracpq 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 : K \leq \frac 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: : K \leq \frac(ab+ac+ad+bc+bd+cd)- \frac(a^2+b^2+c^2+d^2) with equality only for a square.


Diagonals and bimedians

A corollary to Euler's quadrilateral theorem is the inequality : a^2 + b^2 + c^2 + d^2 \ge p^2 + q^2 where equality holds if and only if the quadrilateral is a
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 eq ...
.
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
also generalized
Ptolemy's theorem In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematici ...
, which is an equality in a cyclic quadrilateral, into an inequality for a convex quadrilateral. It states that : pq \le ac + bd where there is equality
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
the quadrilateral is cyclic. This is often called
Ptolemy's inequality In Euclidean geometry, Ptolemy's inequality relates the six distances determined by four points in the plane or in a higher-dimensional space. It states that, for any four points , , , and , the following inequality holds: :\overline\cdot \over ...
. In any convex quadrilateral the bimedians ''m, n'' and the diagonals ''p, q'' are related by the inequality :pq \leq m^2+n^2, with equality holding if and only if the diagonals are equal. This follows directly from the quadrilateral identity m^2+n^2=\tfrac(p^2+q^2).


Sides

The sides ''a'', ''b'', ''c'', and ''d'' of any quadrilateral satisfy :a^2+b^2+c^2 > \frac and :a^4+b^4+c^4 \geq \frac.


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 :K\le \tfracL^2 where ''K'' is the area of a convex quadrilateral with perimeter ''L''. Equality holds
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
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 In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
area is the cyclic quadrilateral. Of all convex quadrilaterals with given diagonals, the orthodiagonal quadrilateral has the largest area. This is a direct consequence of the fact that the area of a convex quadrilateral satisfies :K=\tfracpq\sin\le \tfracpq, 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 :AP+BP+CP+DP\ge AC+BD. 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.


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 In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
(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 mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
s 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''.. In a general convex quadrilateral ''ABCD'', there are no natural analogies to the circumcenter and orthocenter of a
triangle 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 \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
. 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. 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'' = 2''GO''. There can also be defined a ''quasinine-point center'' ''E'' 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''. 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. 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 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

*Let exterior squares be drawn on all sides of a quadrilateral. The segments connecting the centers of opposite squares are (a) equal in length, and (b)
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
. Thus these centers are the vertices of an orthodiagonal quadrilateral. This is called
Van Aubel's theorem In plane geometry, Van Aubel's theorem describes a relationship between squares constructed on the sides of a quadrilateral. Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's ...
. *For any simple quadrilateral with given edge lengths, there is a cyclic quadrilateral with the same edge lengths.Peter, Thomas, "Maximizing the Area of a Quadrilateral", ''The College Mathematics Journal'', Vol. 34, No. 4 (September 2003), pp. 315–316. *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.


Taxonomy

: A hierarchical
taxonomy Taxonomy is the practice and science of categorization or classification. A taxonomy (or taxonomical classification) is a scheme of classification, especially a hierarchical classification, in which things are organized into groups or types. ...
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

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 Cyclobutane is a cycloalkane and organic compound with the formula (CH2)4. Cyclobutane is a colourless gas and commercially available as a liquefied gas. Derivatives of cyclobutane are called cyclobutanes. Cyclobutane itself is of no commerci ...
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 In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
, and conversely every skew quadrilateral comes from a tetrahedron where a pair of opposite
edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed ...
s is removed.


See also

* Complete quadrangle *
Perpendicular bisector construction of a quadrilateral In geometry, the perpendicular bisector construction of a quadrilateral is a construction which produces a new quadrilateral from a given quadrilateral using the perpendicular bisectors to the sides of the former quadrilateral. This construction ari ...
* Saccheri quadrilateral * *
Quadrangle (geography) A "quadrangle" is a topographic map produced by the United States Geological Survey (USGS) covering the United States. The maps are usually named after local physiographic features. The shorthand "quad" is also used, especially with the name of t ...


References


External links

*
Quadrilaterals Formed by Perpendicular BisectorsProjective Collinearity
and
Interactive Classification
of Quadrilaterals from
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Definitions and examples of quadrilaterals
an

from Mathopenref

a


An extended classification of quadrilaterals
a


The role and function of a hierarchical classification of quadrilaterals
by Michael de Villiers {{Polygons 4 (number)