Quadrilateral
{{#invoke:Hatnote|hatnote}} Template:Infobox Polygon In Euclidean plane geometry, a quadrilateral is a polygon with four sides (or edges) 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°. In a crossed quadrilateral, the four interior angles on either side of the crossing add up to 720°.^{[1]}
All convex quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.
{{safesubst:#invoke:anchor|main}}Convex quadrilaterals – parallelograms
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."^{[2]}
- 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).^{[3]}
{{safesubst:#invoke:anchor|main}}Convex quadrilaterals – other
- 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.
- Right kite: a kite with two opposite right angles.
- Trapezoid (North American English) or Trapezium (British English): at least one pair of opposite sides are parallel.
- Trapezium (NAm.): no sides are parallel. (In British English this would be called an irregular quadrilateral, and was once called a trapezoid.)
- Isosceles trapezoid (NAm.) or isosceles trapezium (Brit.): 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.
- Tangential trapezoid: a trapezoid where the four sides are tangents to an inscribed circle.
- 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.
- Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.
- Bicentric quadrilateral: it is both tangential and cyclic.
- Orthodiagonal quadrilateral: the diagonals cross at right angles.
- Equidiagonal quadrilateral: the diagonals are of equal length.
- Ex-tangential quadrilateral: the four extensions of the sides are tangent to an excircle.
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.^{[4]}
- A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square.^{[5]}
- A geometric chevron (dart or arrowhead) is a concave quadrilateral with bilateral symmetry like a kite, but one interior angle is reflex.
- A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed quadrilateral, butterfly quadrilateral or bow-tie quadrilateral. A special case of crossed quadrilaterals are the antiparallelograms, crossed quadrilaterals in which (like a parallelogram) each pair of nonadjacent sides has equal length. The diagonals of a crossed or concave quadrilateral do not intersect inside the shape.
- 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.^{[6]} See skew polygon for more. Historically the term gauche quadrilateral was also used to mean a skew quadrilateral.^{[7]} 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.^{[8]} 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.^{[9]} .
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.^{[10]} 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^{[11]} 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°:^{[12]}
In the case of a parallelogram, the latter formula becomes
Another area formula including the sides a, b, c, d is^{[13]}
where x is the distance between the midpoints of the diagonals and φ is the angle between the bimedians.
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:
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 ^{[18]}^{:p. 126} The corresponding expressions are:{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }}
if the lengths of two bimedians and one diagonal are given, and{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }}
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 (x_{1},y_{1}) and BD as (x_{2},y_{2}), 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.^{[19]} 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^{[20]}
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.^{[18]}^{: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^{[21]}
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^{[22]}^{: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.^{[23]}
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^{[24]} by the Cayley-Menger determinant, as follows:
Bimedians
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.^{[25]}
- 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.^{[18]}^{: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.^{[26]} The length of the bimedian that connects the midpoints of the sides b and d is
Hence^{[18]}^{: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^{[17]}
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:^{[22]}
- 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:^{[27]}
and
Also,^{[28]}
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^{[29]}
- 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.^{[13]}
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^{[30]}
Denoting the perimeter as L, we have^{[30]}^{: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.^{[18]}^{: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.^{[31]}^{:Prop.1} This follows directly from the quadrilateral identity
Sides
The sides a, b, c, and d of any quadrilateral satisfy^{[32]}^{:p.228,#275}
and^{[32]}^{: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^{[30]}^{: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.^{[33]}
Of all convex quadrilaterals with given diagonals, the orthodiagonal quadrilateral has the largest area.^{[30]}^{: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.^{[34]}^{: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.^{[35]}
The "vertex centroid" is the intersection of the two bimedians.^{[36]} 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_{a}G_{c} and G_{b}G_{d}.^{[37]}
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_{a}O_{c} and O_{b}O_{d} is called the quasicircumcenter; and the intersection of the lines H_{a}H_{c} and H_{b}H_{d} is called the quasiorthocenter of the convex quadrilateral.^{[37]} 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.^{[37]}
There can also be defined a quasinine-point center E as the intersection of the lines E_{a}E_{c} and E_{b}E_{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.^{[37]}
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^{[18]}^{: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.^{[33]}
- 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.^{[38]}
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), and "kite" excludes the concave kite (arrowhead or dart). Inclusive definitions are used throughout.
See also
References
- ↑ Stars: A Second Look
- ↑ http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf
- ↑ http://www.cleavebooks.co.uk/scol/calrect.htm
- ↑ G. Keady, P. Scales and S. Z. Németh, "Watt Linkages and Quadrilaterals", The Mathematical Gazette Vol. 88, No. 513 (Nov., 2004), pp. 475–492.
- ↑ A. K. Jobbings, "Quadric Quadrilaterals", The Mathematical Gazette Vol. 81, No. 491 (Jul., 1997), pp. 220–224.
- ↑ M.P. Barnett and J.F. Capitani, Modular chemical geometry and symbolic calculation, International Journal of Quantum Chemistry, 106 (1) 215–227, 2006.
- ↑ William Rowan Hamilton, On Some Results Obtained by the Quaternion Analysis Respecting the Inscription of "Gauche" Polygons in Surfaces of the Second Order, Proceedings of the Royal Irish Academy, 4 (1850), pp. 380–387.
- ↑ Template:Cite web
- ↑ Template:Cite web
- ↑ Harries, J. "Area of a quadrilateral," Mathematical Gazette 86, July 2002, 310–311.
- ↑ R. A. Johnson, Advanced Euclidean Geometry, 2007, Dover Publ., p. 82.
- ↑ Mitchell, Douglas W., "The area of a quadrilateral," Mathematical Gazette 93, July 2009, 306–309.
- ↑ ^{13.0} ^{13.1} {{#invoke:citation/CS1|citation |CitationClass=citation }}.
- ↑ J. L. Coolidge, "A historically interesting formula for the area of a quadrilateral", American Mathematical Monthly, 46 (1939) 345–347.
- ↑ Template:Cite web
- ↑ Archibald, R. C., "The Area of a Quadrilateral", American Mathematical Monthly, 29 (1922) pp. 29–36.
- ↑ ^{17.0} ^{17.1} {{#invoke:citation/CS1|citation |CitationClass=citation }}.
- ↑ ^{18.0} ^{18.1} ^{18.2} ^{18.3} ^{18.4} ^{18.5} Altshiller-Court, Nathan, College Geometry, Dover Publ., 2007.
- ↑ Jennifer Kahle, Geometry: Basic ideas, [1], accessed 28 December 2012.
- ↑ 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.
- ↑ Andreescu, Titu & Andrica, Dorian, Complex Numbers from A to...Z, Birkhäuser, 2006, pp. 207–209.
- ↑ ^{22.0} ^{22.1} {{#invoke:citation/CS1|citation |CitationClass=citation }}.
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}.
- ↑ Template:Cite web
- ↑ H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, MAA, 1967, pp. 52-53.
- ↑ Mateescu Constantin, Answer to Inequality Of Diagonal
- ↑ C. V. Durell & A. Robson, Advanced Trigonometry, Dover, 2003, p. 267.
- ↑ MathPro Press, "Original Problems Proposed by Stanley Rabinowitz 1963–2005", p. 23, [2]
- ↑ O. Bottema, Geometric Inequalities, Wolters–Noordhoff Publishing, The Netherlands, 1969, pp. 129, 132.
- ↑ ^{30.0} ^{30.1} ^{30.2} ^{30.3} {{#invoke:citation/CS1|citation |CitationClass=citation }}.
- ↑ Josefsson, Martin, "Properties of equidiagonal quadrilaterals", Forum Geometricorum, 14 (2014), 129-144. [3]
- ↑ ^{32.0} ^{32.1} Inequalities proposed in “Crux Mathematicorum”, [4].
- ↑ ^{33.0} ^{33.1} Thomas Peter, "Maximizing the Area of a Quadrilateral", The College Mathematics Journal, Vol. 34, No. 4 (September 2003), pp. 315–316.
- ↑ Alsina, Claudi and Nelsen, Roger, Charming Proofs. A Journey Into Elegant Mathematics, Mathematical Association of America, 2010, pp. 114, 119, 120, 261.
- ↑ King, James, Two Centers of Mass of a Quadrilateral, [5], Accessed 2012-04-15.
- ↑ Honsberger, Ross, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Math. Assoc. Amer., 1995, pp. 35–41.
- ↑ ^{37.0} ^{37.1} ^{37.2} ^{37.3} {{#invoke:citation/CS1|citation |CitationClass=citation }}.
- ↑ Martin Josefsson, "Characterizations of Trapezoids", Forum Geometricorum 13 (2013) 23–35.
External links
- Weisstein, Eric W., "Quadrilateral", MathWorld.
- Encyclopedia of Quadri-Figures by Chris Van Tienhoven
- Compendium Geometry Analytic Geometry of Quadrilaterals
- Quadrilaterals Formed by Perpendicular Bisectors, Projective Collinearity and Interactive Classification of Quadrilaterals from cut-the-knot
- Definitions and examples of quadrilaterals and Definition and properties of tetragons from Mathopenref
- A (dynamic) Hierarchical Quadrilateral Tree at Dynamic Geometry Sketches
- An extended classification of quadrilaterals at Dynamic Math Learning Homepage
- Quadrilateral Venn Diagram Quadrilaterals expressed in the form of a Venn diagram, where the areas are also the shape of the quadrilateral they describe.
- The role and function of a hierarchical classification of quadrilaterals by Michael de Villiers
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