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Quadrilateral
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==Simple quadrilaterals== Any quadrilateral that is not self-intersecting is a simple quadrilateral. ===Convex quadrilateral=== [[File:Euler diagram of quadrilateral types.svg|thumb|300px|[[Euler diagram]] of some types of simple quadrilaterals. (UK) denotes British English and (US) denotes American English.]] [[File:Symmetries_of_square.svg|300px|thumb|Convex quadrilaterals by symmetry, represented with a [[Hasse diagram]].]] In a convex quadrilateral all interior angles are less than 180°, and the two diagonals both lie inside the quadrilateral. *Irregular quadrilateral ([[British English]]) or trapezium ([[North American English]]): no sides are parallel. (In British English, this was once called a ''trapezoid''. For more, see {{Section link|Trapezoid|Trapezium vs Trapezoid}}.) *[[Trapezoid|Trapezium]] (UK) or [[trapezoid]] (US): at least one pair of opposite sides are [[parallel (geometry)|parallel]]. Trapezia (UK) and trapezoids (US) include parallelograms. <!--Please do NOT define an isosceles trapezoid as having legs equal. Doing so would make all parallelograms isosceles trapezoids, which we know is wrong.--> *[[Isosceles trapezium]] (UK) or [[isosceles trapezoid]] (US): one pair of opposite sides are parallel and the base [[angle]]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]]: 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 [[Angle#Types of angles|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>{{cite web|url=http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf |title=Archived copy |access-date=June 20, 2013 |url-status=dead |archive-url=https://web.archive.org/web/20140514200449/http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf |archive-date=May 14, 2014 }}</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 (geometry)|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). *[[wikt:oblong|Oblong]]: longer than wide, or wider than long (i.e., a rectangle that is not a square).<ref>{{Cite web|url=http://www.cleavebooks.co.uk/scol/calrect.htm|title=Rectangles Calculator|website=Cleavebooks.co.uk|access-date=1 March 2022}}</ref> *[[Kite (geometry)|Kite]]: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into [[congruent triangles]], and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular. Kites include rhombi. [[File:Quadrilaterals.svg]] *[[Tangential quadrilateral]]: the four sides are tangents to an inscribed circle. A convex quadrilateral is tangential if and only if opposite sides have equal sums. *[[Tangential trapezoid]]: a trapezoid where the four sides are [[tangent]]s 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 angle]]s. *[[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>{{cite journal |first1=G. |last1=Keady |first2=P. |last2=Scales |first3=S. Z. |last3=Németh |title=Watt Linkages and Quadrilaterals |journal=[[The Mathematical Gazette]] |volume=88 |issue=513 |year=2004 |pages=475–492 |doi=10.1017/S0025557200176107 |s2cid=125102050 |url=http://www.m-a.org.uk/jsp/index.jsp?lnk=620 }}</ref> *A ''quadric quadrilateral'' is a convex quadrilateral whose four vertices all lie on the perimeter of a square.<ref>{{cite journal |first=A. K. |last=Jobbings |title=Quadric Quadrilaterals |journal=The Mathematical Gazette |volume=81 |issue=491 |year=1997 |pages=220–224 |doi=10.2307/3619199 |jstor=3619199 |s2cid=250440553 }}</ref> *A ''diametric quadrilateral'' is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle.<ref>{{cite journal |first=R. A. |last=Beauregard |title=Diametric Quadrilaterals with Two Equal Sides |journal=College Mathematics Journal |volume=40 |issue=1 |year=2009 |pages=17–21 |doi=10.1080/07468342.2009.11922331 |s2cid=122206817 }}</ref> *A ''Hjelmslev quadrilateral'' is a quadrilateral with two right angles at opposite vertices.<ref>{{cite book |first=R. |last=Hartshorne |title=Geometry: Euclid and Beyond |publisher=Springer |year=2005 |pages=429–430 |isbn=978-1-4419-3145-0 }}</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 polygon|concave]] quadrilateral with bilateral symmetry like a kite, but where one interior angle is reflex. See [[kite (geometry)|Kite]].
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