Quadrilateral

In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on.

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

Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave.

The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is

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

All convex quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.

    Convex quadrilaterals – parallelograms

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

    A parallelogram is a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms also include the square, rectangle, rhombus and rhomboid.

    • Rhombus or rhomb: all four sides are of equal length. An equivalent condition is that the diagonals perpendicularly bisect each other. An informal description is "a pushed-over square" (including a square).
    • Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles). Informally: "a pushed-over rectangle with no right angles."
    • Rectangle: all four angles are right angles. An equivalent condition is that the diagonals bisect each other and are equal in length. 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), 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 (four equal sides and four equal angles).
    • Oblong: a term sometimes used to denote a rectangle which has unequal adjacent sides (i.e. a rectangle that is not a square).

    Convex quadrilaterals – other

    Self-intersecting quadrilaterals

    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 add up to 720°.

    A special case of crossed quadrilaterals are the antiparallelograms, crossed quadrilaterals in which (like a parallelogram) each pair of nonadjacent sides has equal length.


    More quadrilaterals

    • 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 geometric chevron (dart or arrowhead) is a concave quadrilateral with bilateral symmetry like a kite, but one interior angle is reflex.
    • 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. See skew polygon for more. 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.

    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 Remarkable points 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 K of a convex quadrilateral.

    Trigonometric formulas

    The area can be expressed in trigonometric terms as

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

    The area can be also expressed in terms of bimedians as

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

    Bretschneider's formula expresses the area in terms of the sides and two opposite angles:

    where the sides in sequence are a, b, c, d, where s is the semiperimeter, and A and C are two (in fact, any two) opposite angles. This reduces to Brahmagupta's formula for the area of a cyclic quadrilateral when A+C = 180°.

    Another area formula in terms of the sides and angles, with angle C being between sides b and c, and A being between sides a and d, is

    In the case of a cyclic quadrilateral, the latter formula becomes

    In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to

    Alternatively, we can write the area in terms of the sides and the intersection angle θ of the diagonals, so long as this angle is not 90°:

    In the case of a parallelogram, the latter formula becomes

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

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

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

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

    Non-trigonometric formulas

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

    The first reduces to Brahmagupta's formula in the cyclic quadrilateral case, since then pq = ac + bd.

    The area can also be expressed in terms of the bimedians m, n and the diagonals p, q:

    :Thm. 7

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

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

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

    Vector formulas

    The area of a quadrilateral ABCD can be calculated using vectors. Let vectors AC and BD form the diagonals from A to C and from B to D. The area of the quadrilateral is then

    which is half the magnitude of the cross product of vectors AC and BD. In two-dimensional Euclidean space, expressing vector AC as a free vector in Cartesian space equal to (x1,y1) and BD as (x2,y2), this can be rewritten as:

    Diagonals

    Properties of the diagonals in some quadrilaterals

    In the following table it is listed if the diagonals in some of the most basic quadrilaterals bisect each other, if their diagonals are perpendicular, and if their diagonals have equal length. The list applies to the most general cases, and excludes named subsets.

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

    Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of (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).

    Length of the diagonals

    The length of the diagonals in a convex quadrilateral ABCD can be calculated using the law of cosines. Thus

    and

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

    and

    Generalizations of the parallelogram law and Ptolemy's theorem

    In any convex quadrilateral ABCD, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals plus four times the square of the line segment connecting the midpoints of the diagonals. Thus

    where x is the distance between the midpoints of the diagonals.:p.126 This is sometimes known as Euler's quadrilateral theorem and is a generalization of the parallelogram law.

    The German mathematician Carl Anton Bretschneider derived in 1842 the following generalization of Ptolemy's theorem, regarding the product of the diagonals in a convex quadrilateral

    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:p.14

    In a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD, d = DA, and where the diagonals intersect at E,

    where e = AE, f = BE, g = CE, and h = DE.

    The shape of a convex quadrilateral is fully determined by the lengths of its sides in sequence and of one diagonal between two specified vertices. The two diagonals p, q and the four side lengths a, b, c, d of a quadrilateral are related by the Cayley-Menger determinant, as follows:

    Bimedians

    The Varignon parallelogram EFGH

    The bimedians of a quadrilateral are the line segments connecting the midpoints of the opposite sides. 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.
    • 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.:p.125

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

    where p and q are the length of the diagonals. The length of the bimedian that connects the midpoints of the sides b and d is

    Hence:p.126

    This is also a corollary to the parallelogram law applied in the Varignon parallelogram.

    The lengths of the bimedians can also be expressed in terms of two opposite sides and the distance x between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence

    and

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

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

    • 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:

    and

    Also,

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

    Inequalities

    Area

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

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

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

    with equality if and only if the quadrilateral is cyclic or degenerate such that one side is equal to the sum of the other three (it has collapsed into a line segment, so the area is zero).

    The area of any quadrilateral also satisfies the inequality

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

    with equality only in the case of a square.

    The area of a convex quadrilateral also satisfies

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

    Diagonals and bimedians

    A corollary to Euler's quadrilateral theorem is the inequality

    where equality holds if and only if the quadrilateral is a parallelogram.

    Euler also generalized Ptolemy's theorem, which is an equality in a cyclic quadrilateral, into an inequality for a convex quadrilateral. It states that

    where there is equality if and only if the quadrilateral is cyclic.:p.128–129 This is often called Ptolemy's inequality.

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

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

    Sides

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

    and:p.234,#466

    Maximum and minimum properties

    Among all quadrilaterals with a given perimeter, the one with the largest area is the square. This is called the isoperimetric theorem for quadrilaterals. It is a direct consequence of the area inequality:p.114

    where K is the area of a convex quadrilateral with perimeter L. Equality holds if and only if the quadrilateral is a square. The dual theorem states that of all quadrilaterals with a given area, the square has the shortest perimeter.

    The quadrilateral with given side lengths that has the maximum area is the cyclic quadrilateral.

    Of all convex quadrilaterals with given diagonals, the orthodiagonal quadrilateral has the largest area.:p.119 This is a direct consequence of the fact that the area of a convex quadrilateral satisfies

    where θ is the angle between the diagonals p and q. Equality holds if and only if θ = 90°.

    If P is an interior point in a convex quadrilateral ABCD, then

    From this inequality it follows that the point inside a quadrilateral that minimizes the sum of distances to the vertices is the intersection of the diagonals. Hence that point is the Fermat point of a convex quadrilateral.:p.120

    Remarkable points and lines in a convex quadrilateral

    The centre of a quadrilateral can be defined in several different ways. The "vertex centroid" comes from considering the quadrilateral as being empty but having equal masses at its vertices. The "side centroid" comes from considering the sides to have constant mass per unit length. The usual centre, called just centroid (centre of area) comes from considering the surface of the quadrilateral as having constant density. These three points are in general not all the same point.

    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 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. 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 = 2GO.

    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 quadrilateral is the Newton line.

    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. Thus these centers are the vertices of an orthodiagonal quadrilateral. This is called Van Aubel's theorem.
    • The internal angle bisectors of a convex quadrilateral either form a cyclic quadrilateral:p.127 or they are concurrent. In the latter case the quadrilateral is a tangential quadrilateral.
    • 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.
    • 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.

    Taxonomy

    A taxonomy of quadrilaterals is illustrated by the following graph. Lower forms are special cases of higher forms. Note that "trapezoid" here is referring to the North American definition (the British equivalent is a trapezium). Inclusive definitions are used throughout.

    Taxonomic hierarchy of quadrilaterals. Lower forms are special cases of the higher forms they are connected to.

    See also

    References

    External links

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