Трапез (геометрија)

Трапез
Трапез
	Трапез
Тип	четвороугао
Ивице и темена	4
Површина
Својства	конвексан

Шаблон:Short description

У Еуклидовој геометрији, трапез је конвексни четвороугао са најмање једним паром паралелних страница.^[1]^[2] Паралелне странице се зову основице трапеза, а друге две странице се зову краци или бочне странице (ако нису паралелне, онда постоје два пара основица). Скаленски трапез је трапез без икаквих страница са једнаким мерама,^[3] за разлику од посебних случајева испод.

Висина трапеза h је растојање између две паралелне странице. Збир углова на једном од кракова је 180° тј. α + δ = β + γ = 180°.

Етимологија трапез vs трапезоид

Ancient Greek mathematician Euclid defined five types of quadrilateral, of which four had two sets of parallel sides (known in English as square, rectangle, rhombus and rhomboid) and the last did not have two sets of parallel sides – a τραπέζια (trapezia^[5] literally "a table", itself from τετράς (tetrás), "four" + πέζα (péza), "a foot; end, border, edge").^[6]

Two types of trapezia were introduced by Proclus (412 to 485 AD) in his commentary on the first book of Euclid's Elements:^[4]^[7]

one pair of parallel sides – a trapezium (τραπέζιον), divided into isosceles (equal legs) and scalene (unequal) trapeziums
no parallel sides – trapezoid (τραπεζοειδή, trapezoeidé, literally trapezium-like (εἶδος means "resembles"), in the same way as cuboid means cube-like and rhomboid means rhombus-like)

All European languages follow Proclus's structure^[7]^[8] as did English until the late 18th century, until an influential mathematical dictionary published by Charles Hutton in 1795 supported without explanation a transposition of the terms. This mistake was corrected in British English in about 1875, but was retained in American English into the modern day.^[4]

Type	Original terminology			Modern terminology
	Euclid (Definition 22)	Proclus (Definitions 30-34, quoting Posidonius)	Euclid / Proclus definition	British English (and European languages)	American English
Parallelogram	ῥόμβος (rhombos)		equilateral but not right-angled	Rhombus		Trapezoid (inclusive)
Parallelogram	ῥομβοειδὲς (rhomboides)		opposite sides and angles equal to one another but not equilateral nor right-angled	Rhomboid (colloquially Parallelogram)
Non-parallelogram	τραπέζια (trapezia)	τραπέζιον ἰσοσκελὲς (trapezion isoskelés)	Two parallel sides, and a line of symmetry	Isoceles Trapezium	Isoceles Trapezoid
		τραπέζιον σκαληνὸν (trapezion skalinón)	Two parallel sides, and no line of symmetry	Trapezium	Trapezoid (exclusive)
		τραπέζοειδὲς (trapezoides)	No parallel sides	Trapezoid	Trapezium

The shape is often called an irregular quadrilateral.^[9]^[10]

Инклузивна наспрам ексклузивне дефиниције

Постоје одређена неслагања да ли паралелограме, који имају два пара паралелних страница, треба сматрати трапезоидима. Неки дефинишу трапез као четвороугао који има само један пар паралелних страница (ексклузивна дефиниција), чиме се искључују паралелограми.^[11] Други^[12] дефинишу трапез као четвороугао са најмање једним паром паралелних страница (инклузивна дефиниција^[13]), чинећи паралелограм посебним типом трапеза. Последња дефиниција је у складу са њеном употребом у вишој математици као што је калкулус. Овај чланак користи инклузивну дефиницију и разматра паралелограме као посебне случајеве трапеза. Ово се заговара и у таксономији четвороуглова.

Посебни случајеви

Посебни случајеви трапеза су:

једнакокраки трапез, код кога су краци једнаки, такође и углови на основици су једнаки
правоугли трапез, код кога је један крак управан на базу, тада је тај крак истовремено и висина
паралелограм, код кога је и други пар страница међусобно паралелан
ромб, који је паралелограм, али су му и све странице међусобно једнаке
правоугаоник, који је паралелограм, али су му и све суседне странице међусобно нормалне
квадрат, коме су све странице међусобно једнаке, а суседне међусобно нормалне

Услов постојања

Four lengths a, c, b, d can constitute the consecutive sides of a non-parallelogram trapezoid with a and b parallel only when^[14]

\displaystyle |d-c|<|b-a|<d+c.

The quadrilateral is a parallelogram when $d-c=b-a=0$ , but it is an ex-tangential quadrilateral (which is not a trapezoid) when $|d-c|=|b-a|\neq 0$ .^[15]^{‍:p. 35}

Карактеризације

Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:

It has two adjacent angles that are supplementary, that is, they add up to 180 degrees.
The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal.
The diagonals cut each other in mutually the same ratio (this ratio is the same as that between the lengths of the parallel sides).
The diagonals cut the quadrilateral into four triangles of which one opposite pair have equal areas.^[15]^‍:Prop.5
The product of the areas of the two triangles formed by one diagonal equals the product of the areas of the two triangles formed by the other diagonal.^[15]^‍:Thm.6
The areas S and T of some two opposite triangles of the four triangles formed by the diagonals satisfy the equation

{\sqrt {K}}={\sqrt {S}}+{\sqrt {T}},

where K is the area of the quadrilateral.^[15]^‍:Thm.8

The midpoints of two opposite sides and the intersection of the diagonals are collinear.^[15]^‍:Thm.15
The angles in the quadrilateral ABCD satisfy $\sin A\sin C=\sin B\sin D.$ ^[15]^{‍:p. 25}
The cosines of two adjacent angles sum to 0, as do the cosines of the other two angles.^[15]^{‍:p. 25}
The cotangents of two adjacent angles sum to 0, as do the cotangents of the other two adjacent angles.^[15]^{‍:p. 26}
One bimedian divides the quadrilateral into two quadrilaterals of equal areas.^[15]^{‍:p. 26}
Twice the length of the bimedian connecting the midpoints of two opposite sides equals the sum of the lengths of the other sides.^[15]^{‍:p. 31}

Additionally, the following properties are equivalent, and each implies that opposite sides a and b are parallel:

The consecutive sides a, c, b, d and the diagonals p, q satisfy the equation^[15]^‍:Cor.11

p^{2}+q^{2}=c^{2}+d^{2}+2ab.

The distance v between the midpoints of the diagonals satisfies the equation^[15]^‍:Thm.12

v={\frac {|a-b|}{2}}.

Формуле

Обим	$O=a+b+c+d\,$
Висина	$h=b\sin {\beta }=d\sin {\alpha }\,$ $h\,=\,{\frac {2}{a-c}}{\sqrt {s(s-(a-c))(s-b)(s-d)}},\;s\,=\,{\frac {a+c+b+d}{2}}$
Површина	$P\,=\,{\frac {a+b}{2}}\cdot h$
Дијагонале	$d_{1}\,=\,{\sqrt {a^{2}+b^{2}-2\,a\,b\,\cos \beta }}={\sqrt {c^{2}+d^{2}-2\,c\,d\,\cos \delta }}$ $d_{2}\,=\,{\sqrt {a^{2}+d^{2}-2\,a\,d\,\cos \alpha }}={\sqrt {b^{2}+c^{2}-2\,b\,c\,\cos \gamma }}$

Једнакокраки трапез

Код једнакокраког трапеза важи да је b = d, такође је α = β одакле следи δ = γ. Последица овога је да је збир наспрамних углова α + γ = β + δ = 180°. Ово је особина тетивних четвороуглова, значи једнакокраки трапез је тетивни четвороугао.

Правоугли трапез

Код правоуглог трапеза је b или d једнако h, а такође важи да је α = δ = 90° ili β = γ = 90°.

Референце

^ „Trapezoid — Math Word Definition”. Math Open Reference. Приступљено 9. 9. 2018.
^ A. D. Gardiner & C. J. Bradley, Plane Euclidean Geometry: Theory and Problems, UKMT, 2005, p. 34.
^ „Types of quadrilaterals”. Basic-mathematics.com. Приступљено 2018-09-09.
^ ^а ^б ^в James A. H. Murray (1926). A New English Dictionary on Historical Principles: Founded Mainly on the Materials Collected by the Philological Society. X. Clarendon Press at Oxford. стр. 286 (Trapezium). „With Euclid (c 300 B.C.) τραπέζιον included all quadrilateral figures except the square, rectangle, rhombus, and rhomboid; into the varieties of trapezia he did not enter. But Proclus, who wrote Commentaries on the First Book of Euclid's Elements A.D. 450, retained the name τραπέζιον only for quadrilaterals having two sides parallel, subdividing these into the τραπέζιον ἰσοσκελὲς, isosceles trapezium, having the two non-parallel sides (and the angles at their bases) equal, and σκαληνὸν τραπέζιον, scalene trapezium, in which these sides and angles are unequal. For quadrilaterals having no sides parallel, Proclus introduced the name τραπέζοειδὲς TRAPEZOID. This nomenclature is retained in all the continental languages, and was universal in England till late in the 18th century, when the application of the terms was transposed, so that the figure which Proclus and modern geometers of other nations call specifically a trapezium (F. trapèze, Ger. trapez, Du. trapezium, It. trapezio) became with most English writers a trapezoid, and the trapezoid of Proclus and other nations a trapezium. This changed sense of trapezoid is given in Hutton's Mathematical Dictionary, 1795, as ‘sometimes’ used -- he does not say by whom; but he himself unfortunately adopted and used it, and his Dictionary was doubtless the chief agent in its diffusion. Some geometers however continued to use the terms in their original senses, and since c 1875 this is the prevalent use.”
^ Euclid Elements Book I Definition 22
^ πέζα is said to be the Doric and Arcadic form of πούς "foot", but recorded only in the sense "instep [of a human foot]", whence the meaning "edge, border". τράπεζα "table" is Homeric. Henry George Liddell, Robert Scott, Henry Stuart Jones, A Greek-English Lexicon, Oxford, Clarendon Press (1940), s.v. πέζα, τράπεζα.
^ ^а ^б Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (5. 4. 2016). The Symmetries of Things. CRC Press. стр. 286. ISBN 978-1-4398-6489-0.
^ For example: French trapèze, Italian trapezio, Portuguese trapézio, Spanish trapecio, German Trapez, Ukrainian "трапеція", e.g. „Larousse definition for trapézoïde”.
^ Chambers 21st Century Dictionary Trapezoid
^ „1913 American definition of trapezium”. Merriam-Webster Online Dictionary. Приступљено 2007-12-10.
^ „American School definition from "math.com"”. Приступљено 2008-04-14.
^ Weisstein, Eric W. „Trapezoid”. MathWorld.
^ Trapezoids, [1]. Retrieved 2012-02-24.
^ Ask Dr. Math (2008), "Area of Trapezoid Given Only the Side Lengths".
^ ^а ^б ^в ^г ^д ^ђ ^е ^ж ^з ^и ^ј ^к Martin Josefsson, "Characterizations of trapezoids", Forum Geometricorum, 13 (2013) 23-35.

Литература

D. Fraivert, A. Sigler and M. Stupel : Common properties of trapezoids and convex quadrilaterals
„Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram”. Mathsisfun.com. Приступљено 2020-09-02.
Martin, George Edward (1982), Transformation geometry, Undergraduate Texts in Mathematics, Springer-Verlag, Theorem 12.1, page 120, ISBN 0-387-90636-3, MR 718119, doi:10.1007/978-1-4612-5680-9
„Archived copy” (PDF). Архивирано из оригинала (PDF) 14. 5. 2014. г. Приступљено 20. 6. 2013.
Keady, G.; Scales, P.; Németh, S. Z. (2004). „Watt Linkages and Quadrilaterals”. The Mathematical Gazette. 88 (513): 475—492. S2CID 125102050. doi:10.1017/S0025557200176107.
Jobbings, A. K. (1997). „Quadric Quadrilaterals”. The Mathematical Gazette. 81 (491): 220—224. JSTOR 3619199. doi:10.2307/3619199.
Beauregard, R. A. (2009). „Diametric Quadrilaterals with Two Equal Sides”. College Mathematics Journal. 40 (1): 17—21. S2CID 122206817. doi:10.1080/07468342.2009.11922331.
Hartshorne, R. (2005). Geometry: Euclid and Beyond. Springer. стр. 429—430. ISBN 978-1-4419-3145-0.
Josefsson, Martin (2013), „Five Proofs of an Area Characterization of Rectangles” (PDF), Forum Geometricorum, 13: 17—21
Josefsson, Martin (2011), „The Area of a Bicentric Quadrilateral” (PDF), Forum Geometricorum, 11: 155—164
Leonard Mihai Giugiuc; Dao Thanh Oai; Kadir Altintas (2018). „An inequality related to the lengths and area of a convex quadrilateral” (PDF). International Journal of Geometry. 7: 81—86.

Спољашње везе

Trapezium at Encyclopedia of Mathematics.
Weisstein, Eric W. „Right trapezoid”. MathWorld.
Trapezoid definition Area of a trapezoid Median of a trapezoid With interactive animations
Trapezoid (North America) at elsy.at: Animated course (construction, circumference, area)
Trapezoidal Rule on Numerical Methods for Stem Undergraduate
Autar Kaw and E. Eric Kalu, Numerical Methods with Applications, (2008)

[1] „Trapezoid — Math Word Definition”. Math Open Reference. Приступљено 9. 9. 2018.

[2] A. D. Gardiner & C. J. Bradley, Plane Euclidean Geometry: Theory and Problems, UKMT, 2005, p. 34.

[3] „Types of quadrilaterals”. Basic-mathematics.com. Приступљено 2018-09-09.

[oed-4] а ^б ^в James A. H. Murray (1926). A New English Dictionary on Historical Principles: Founded Mainly on the Materials Collected by the Philological Society. X. Clarendon Press at Oxford. стр. 286 (Trapezium). „With Euclid (c 300 B.C.) τραπέζιον included all quadrilateral figures except the square, rectangle, rhombus, and rhomboid; into the varieties of trapezia he did not enter. But Proclus, who wrote Commentaries on the First Book of Euclid's Elements A.D. 450, retained the name τραπέζιον only for quadrilaterals having two sides parallel, subdividing these into the τραπέζιον ἰσοσκελὲς, isosceles trapezium, having the two non-parallel sides (and the angles at their bases) equal, and σκαληνὸν τραπέζιον, scalene trapezium, in which these sides and angles are unequal. For quadrilaterals having no sides parallel, Proclus introduced the name τραπέζοειδὲς TRAPEZOID. This nomenclature is retained in all the continental languages, and was universal in England till late in the 18th century, when the application of the terms was transposed, so that the figure which Proclus and modern geometers of other nations call specifically a trapezium (F. trapèze, Ger. trapez, Du. trapezium, It. trapezio) became with most English writers a trapezoid, and the trapezoid of Proclus and other nations a trapezium. This changed sense of trapezoid is given in Hutton's Mathematical Dictionary, 1795, as ‘sometimes’ used -- he does not say by whom; but he himself unfortunately adopted and used it, and his Dictionary was doubtless the chief agent in its diffusion. Some geometers however continued to use the terms in their original senses, and since c 1875 this is the prevalent use.”

[5] Euclid Elements Book I Definition 22

[6] πέζα is said to be the Doric and Arcadic form of πούς "foot", but recorded only in the sense "instep [of a human foot]", whence the meaning "edge, border". τράπεζα "table" is Homeric. Henry George Liddell, Robert Scott, Henry Stuart Jones, A Greek-English Lexicon, Oxford, Clarendon Press (1940), s.v. πέζα, τράπεζα.

[ConwayBurgiel2016-7] а ^б Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (5. 4. 2016). The Symmetries of Things. CRC Press. стр. 286. ISBN 978-1-4398-6489-0.

[8] For example: French trapèze, Italian trapezio, Portuguese trapézio, Spanish trapecio, German Trapez, Ukrainian "трапеція", e.g. „Larousse definition for trapézoïde”.

[9] Chambers 21st Century Dictionary Trapezoid

[10] „1913 American definition of trapezium”. Merriam-Webster Online Dictionary. Приступљено 2007-12-10.

[11] „American School definition from "math.com"”. Приступљено 2008-04-14.

[Mathworld-12] Weisstein, Eric W. „Trapezoid”. MathWorld.

[13] Trapezoids, [1]. Retrieved 2012-02-24.

[14] Ask Dr. Math (2008), "Area of Trapezoid Given Only the Side Lengths".

[Josefsson-15] а ^б ^в ^г ^д ^ђ ^е ^ж ^з ^и ^ј ^к Martin Josefsson, "Characterizations of trapezoids", Forum Geometricorum, 13 (2013) 23-35.

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