External sorting - Revision history

en>MSUGRA: /* Additional passes */

2014-02-23T14:08:24Z

Additional passes

← Older revision		Revision as of 16:08, 23 February 2014
Line 1:		Line 1:
	~~[[File:Menelaus' theorem 1.svg\|right\|thumb\|Menelaus' theorem, case 1: line DEF passes inside triangle ABC]]~~

	'''Menelaus' theorem''', named for [[Menelaus of Alexandria]], is a theorem about [[triangle]]s in [[plane geometry]]. Given a triangle ''ABC'', and a [[Transversal (geometry)\|transversal]] line that crosses ''BC'', ''AC'' and ''AB'' at points ''D'', ''E'' and ''F'' respectively, with ''D'', ''E'', and ''F'' distinct from ''A'', ''B'' and ''C'', then
	~~: <math>\frac{AF}{FB} \times \frac{BD}{DC} \times \frac{CE}{EA} = -1.</math>~~

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	~~: <math>AF \times BD \times CE=~~ - ~~FB \times DC \times EA .</math>~~

	This equation uses signed lengths of segments, in other words the length ''AB'' is taken to be positive or negative according to whether ''A'' is to the left or right of ''B'' in some fixed orientation of the line. For example, ''AF''/''FB'' is defined as having positive value when ''F'' is between ''A'' and ''B'' and ~~negative otherwise.~~

	~~The~~ [~~[Theorem#Converse\|converse]] is also true~~: ~~If points ''D'', ''E'' and ''F'' are chosen on ''BC'', ''AC'' and ''AB'' respectively so that~~
	~~: <math>\frac{AF}{FB} \times \frac{BD}{DC} \times \frac{CE}{EA} = -1,<~~/~~math>~~
	~~then ''D'', ''E'' and ''F'' are collinear~~. ~~The converse is often included as part of the theorem~~.

	~~The theorem is very similar to [[Ceva's theorem]] in that their equations differ only in sign.~~

	=~~= Proof ==~~
	~~[[File:Menelaos' theorem 2.svg\|right\|thumb\|Menelaus' theorem, case 2: line DEF is entirely outside triangle ABC~~]]

	~~A standard proof is as follows:<ref>Follows Russel</ref>~~

	~~First, the sign of the [[left~~-~~hand side]] will be negative since either all three of the ratios are negative, the case where the~~ line ~~DEF misses the triangle (lower diagram), or one is negative and the other two~~ are ~~positive, the case where DEF crosses two sides~~ of ~~the triangle. (See [[Pasch's axiom]].)~~

	~~To check the magnitude, construct perpendiculars from ''A'', ''B''~~, and ~~''C''~~ to the ~~line ''DEF'' and let their lengths be ''a, b,'' and ''c'' respectively. Then by~~ [~~[Similarity (geometry)\|similar]] triangles it follows that \|''AF''/''FB''\| = \|''a''/''b''\|, \|''BD''~~/~~''DC''\| = \|''b''~~/~~''c''\|, and \|''CE''/''EA''\| = ''c''/''a''~~. So
	~~: <math>\left\|\frac{AF}{FB}\right\| \cdot \left\|\frac{BD}{DC}\right\| \cdot \left\|\frac{CE}{EA}\right\| = \left\| \frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{a} \right\| = 1~~.<~~/math~~>

	~~For a simpler, if less symmetrical way~~ to ~~check~~ the magnitude,<ref>Follows {{cite book \|title=Inductive Plane Geometry\|first=George Irving\|last=Hopkins\|publisher=D.C. Heath & Co.\|year=1902\|chapter=Art. 983}}</ref> draw ''CK'' parallel to ''AB'' where ''DEF'' meets ''CK'' at ''K''. Then by similar triangles
	: ~~<math>\left\|\frac{BD}{DC}\right\| = \left\|\frac{BF}{CK}\right\|,\,\left\|\frac{AE}{EC}\right\| = \left\|\frac{AF}{CK}\right\|<~~/~~math>~~
	~~and the result follows by eliminating ''CK'' from these equations.~~

	~~The converse follows as a corollary.<ref>Follows Russel with some simplification<~~/~~ref> Let ''D'', ''E'' and ''F'' be given on the lines ''BC'', ''AC'' and ''AB'' so that the equation holds~~. ~~Let ''F''′ be the point where ''DE'' crosses ''AB''~~. ~~Then by the theorem, the equation also holds for ''D'', ''E'' and ''F''′~~. ~~Comparing the two,~~
	~~: <math>\frac{AF}{FB} = \frac{AF'}{F'B}.</math>~~
	~~But at most one point can cut a segment in a given ratio so ''F''=''F''′.~~

	~~=== A non-computational proof using homothecies ===~~
	~~The following proof<ref>See Michèle Audin, Géométrie, éditions BELIN, Paris 1998: indication for exercice 1.37, p. 273</ref> uses only notions of [[affine geometry]~~], ~~notably [[Homothetic transformation\|homothecies]].~~
	Whether or not ''D'', ''E'', ''F'' are collinear, there are three homothecies with centers ''D'', ''E'', ''F'' that respectively send ''B'' to ''C'', ''C'' to ''A'', and ''A'' to ''B''. The composition of the three then is an element of the group of homothecy-~~translations that fixes ''B'', so it is~~ a ~~homothecy with center ''B'', possibly with ratio 1 (in which case it is the identity)~~. ~~This composition fixes the line ''DE'' if~~ and only if ''F'' is collinear with ''D'' and ''E'' (since the first two homothecies certainly fix ''DE'', and the third does so only if ''F'' lies on ''DE''). Therefore ''D'', ''E'', ''F'' are collinear if and only if this composition is the identity, which means that the product of the three ratios is 1:
	~~: <math>\frac{\overrightarrow{DC}}{\overrightarrow{DB}} \times~~
	~~\frac{\overrightarrow{EA}}{\overrightarrow{EC}} \times~~
	~~\frac{\overrightarrow{FB}}{\overrightarrow{FA}} = 1,</math>~~
	~~which is equivalent~~ to ~~the given equation.~~

	~~==History==~~
	~~It is uncertain who actually discovered the theorem; however, the oldest extant exposition appears in ''Spherics'' by Menelaus. In this book, the plane version of the theorem is used~~ as a ~~lemma to prove a spherical version of the theorem~~.~~<ref>{{cite book\|~~
	~~last=Smith\|first=D~~.~~E.\|title=History of Mathematics\|isbn=0-486-20430-8~~
	~~\|publisher=Courier Dover Publications\|year=1958\|volume=II\|page=607}}</ref>~~

	In [[Almagest]], [[Ptolemy]] applies the theorem on a number of problems in spherical astronomy.<ref name="rashed">{{cite book\|last=Rashed\|first=Roshdi\|title=Encyclopedia of the history of Arabic science\|volume=2\|page=483\|year=1996\|publisher=Routledge\|location=London\|isbn=0-415-02063-8}}</ref> During the [[Islamic Golden Age]], Muslim scholars devoted a number of works that engaged in the study of Menelaus' theorem, which they referred to as "the proposition on the secants" (''shakl al-qatta'''). The [[complete quadrilateral]] was called the "figure of secants" in their terminology.<ref name="rashed" /> [[Al-Biruni]]'s work, ''The Keys of Astronomy'', lists a number of those works, which can be classified into studies as part of commentaries on Ptolemy's ''Almagest'' as in the works of [[al-Nayrizi]] and [[al-Khazin]] where each demonstrated particular cases of Menelaus' theorem that led to the [[sine rule]],<ref name="musa">{{cite journal\|last=Moussa\|first=Ali\|title=Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations\|journal=Arabic Sciences and Philosophy\|year=2011\|volume=21\|issue=1\|publisher=[[Cambridge University Press]]\|doi=10.~~1017/S095742391000007X}}</ref> or works composed as independent treatises such as:~~

	* The "Treatise on the Figure of Secants" (''Risala fi shakl al-qatta''') by [[Thabit ibn Qurra]].~~<ref name="rashed" />~~
	* [[Husam al-DIn al-Salar]]~~'s ''Removing the Veil from the Mysteries of the Figure of Secants'' (Kashf al-qina' '~~an ~~asrar al-shakl al-qatta')~~, ~~also known as "The Book on~~ the ~~Figure~~ of ~~Secants" (''Kitab al-shakl al-qatta''') or in Europe as ''The Treatise on the Complete Quadrilateral''. The lost treatise was referred to by [[Al-Tusi]] and [[Nasir al-Din al-Tusi]]~~.<~~ref name="rashed" /~~>
	* Work by [[al-Sijzi]].<ref name="musa" />
	* ''Tahdhib'' by [~~[Abu Nasr ibn Iraq]].<ref name="musa" />~~

	~~==References==~~
	~~{{Reflist}}~~
	* {{cite book \|title=Pure Geometry
	~~\|first=John Wellesley\|last=Russell\|publisher=Clarendon Press\|year=1905~~
	~~\|chapter= Ch. 1 §6 "Menelaus' Theorem"~~
	~~\|url=~~http://~~books~~.~~google.com/books?id=r3ILAAAAYAAJ}}~~

	~~== External links ==~~
	* [http://planetmath.org/?~~op=getobj&from~~=~~objects&id=3092 Alternate proof~~] ~~of Menelaus' theorem, from [[PlanetMath]]~~
	* [http://www.~~cut-~~the~~-knot~~.~~org/Curriculum/Geometry/MenelausFromCeva~~.~~shtml Menelaus From Ceva]~~
	* [http://www.~~cut-the-knot~~.~~org/4travelers/CevaAndMenelaus~~.~~shtml Ceva and Menelaus Meet on the Roads]~~
	* [http://www.~~mathpages~~.~~com/home/kmath442/kmath442.htm Menelaus and Ceva] at MathPages~~
	* [http://demonstrations.wolfram.com/MenelausTheorem/ Menelaus' Theorem] by Jay Warendorff. [[The Wolfram Demonstrations Project]].
	* {{MathWorld \|title=Menelaus' Theorem \|urlname=MenelausTheorem}}

	~~[[Category:Affine geometry]]~~
	~~[[Category:Triangle geometry]]~~
	~~[[Category:Articles containing proofs]]~~
	~~[[Category:Theorems in plane geometry]~~]

24.7.64.61: as i understand it, external sorts can be distribution or merge sorts as those terms are usually used

2014-02-01T00:03:02Z

as i understand it, external sorts can be distribution or merge sorts as those terms are usually used

← Older revision		Revision as of 02:03, 1 February 2014
Line 1:		Line 1:
	~~Luke~~ is ~~usually~~ a ~~celebrity during the building as well as occupation progress 1st 2nd to his 3rd club~~ [~~http~~://~~minioasis~~.~~com luke bryan concert tour schedule~~] ~~record~~, & , is ~~definitely~~ the ~~resistant~~. ~~He burst~~ to the ~~picture in 2004 regarding his best combination~~ of ~~down~~-~~residence ease~~ of ~~access~~, ~~video legend good appearance~~ and ~~lines~~, ~~is set t in a significant~~ way. The ~~new album Top about~~ the ~~nation graph~~ and ~~#2 about~~ the ~~pop maps~~, ~~making it~~ the ~~next top very first during that time of 2012~~ for the ~~region performer~~. <br><br>The ~~child~~ of the , is ~~aware persistence and dedication are key elements when~~ it ~~comes to an effective occupation-~~ . ~~His~~ first ~~record~~, ~~Continue to be Me~~, ~~produced~~ the ~~very best~~ ~~reaches “All My Buddies Say” and “Country Man~~,~~” when his effort~~, ~~Doin’ Thing~~, ~~identified~~ the ~~performer~~-~~about three directly No. 2 single people:~~ In ~~addition Contacting Is~~ a ~~Great Factor~~.”<br><br>During the ~~fall~~ of ~~1999~~, ~~Concerts: Bryan And that had~~ [~~http:~~/~~/lukebryantickets.neodga.com the today show luke bryan~~] ~~an impressive set~~ of , ~~such as City. “It’s much like you are obtaining~~ a ~~approval to look to another level~~, ~~claims those artists~~ which ~~were a~~ part of the ~~Tourmore than~~ in [~~http:~~//~~lukebryantickets~~.~~lazintechnologies.com ticket brokers~~] ~~a greater level~~ of ~~musicians.” It packaged among~~ the ~~best trips~~ in ~~its 15~~-~~year historical past~~.<br><br>~~Feel free to visit my blog~~ ... [http://~~lukebryantickets~~.~~iczmpbangladesh~~.org ~~2014 luke bryan concerts~~]		[[File:Menelaus' theorem 1.svg\|right\|thumb\|Menelaus' theorem, case 1: line DEF passes inside triangle ABC]]

			'''Menelaus' theorem''', named for [[Menelaus of Alexandria]], is a theorem about [[triangle]]s in [[plane geometry]]. Given a triangle ''ABC'', and a [[Transversal (geometry)\|transversal]] line that crosses ''BC'', ''AC'' and ''AB'' at points ''D'', ''E'' and ''F'' respectively, with ''D'', ''E'', and ''F'' distinct from ''A'', ''B'' and ''C'', then
			: <math>\frac{AF}{FB} \times \frac{BD}{DC} \times \frac{CE}{EA} = -1.</math>

			or simply

			: <math>AF \times BD \times CE= - FB \times DC \times EA .</math>

			This equation uses signed lengths of segments, in other words the length ''AB'' is taken to be positive or negative according to whether ''A'' is to the left or right of ''B'' in some fixed orientation of the line. For example, ''AF''/''FB'' is defined as having positive value when ''F'' is between ''A'' and ''B'' and negative otherwise.

			The [[Theorem#Converse\|converse]] is also true: If points ''D'', ''E'' and ''F'' are chosen on ''BC'', ''AC'' and ''AB'' respectively so that
			: <math>\frac{AF}{FB} \times \frac{BD}{DC} \times \frac{CE}{EA} = -1,</math>
			then ''D'', ''E'' and ''F'' are collinear. The converse is often included as part of the theorem.

			The theorem is very similar to [[Ceva's theorem]] in that their equations differ only in sign.

			== Proof ==
			[[File:Menelaos' theorem 2.svg\|right\|thumb\|Menelaus' theorem, case 2: line DEF is entirely outside triangle ABC]]

			A standard proof is as follows:<ref>Follows Russel</ref>

			First, the sign of the [[left-hand side]] will be negative since either all three of the ratios are negative, the case where the line DEF misses the triangle (lower diagram), or one is negative and the other two are positive, the case where DEF crosses two sides of the triangle. (See [[Pasch's axiom]].)

			To check the magnitude, construct perpendiculars from ''A'', ''B'', and ''C'' to the line ''DEF'' and let their lengths be ''a, b,'' and ''c'' respectively. Then by [[Similarity (geometry)\|similar]] triangles it follows that \|''AF''/''FB''\| = \|''a''/''b''\|, \|''BD''/''DC''\| = \|''b''/''c''\|, and \|''CE''/''EA''\| = ''c''/''a''. So
			: <math>\left\|\frac{AF}{FB}\right\| \cdot \left\|\frac{BD}{DC}\right\| \cdot \left\|\frac{CE}{EA}\right\| = \left\| \frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{a} \right\| = 1.</math>

			For a simpler, if less symmetrical way to check the magnitude,<ref>Follows {{cite book \|title=Inductive Plane Geometry\|first=George Irving\|last=Hopkins\|publisher=D.C. Heath & Co.\|year=1902\|chapter=Art. 983}}</ref> draw ''CK'' parallel to ''AB'' where ''DEF'' meets ''CK'' at ''K''. Then by similar triangles
			: <math>\left\|\frac{BD}{DC}\right\| = \left\|\frac{BF}{CK}\right\|,\,\left\|\frac{AE}{EC}\right\| = \left\|\frac{AF}{CK}\right\|</math>
			and the result follows by eliminating ''CK'' from these equations.

			The converse follows as a corollary.<ref>Follows Russel with some simplification</ref> Let ''D'', ''E'' and ''F'' be given on the lines ''BC'', ''AC'' and ''AB'' so that the equation holds. Let ''F''′ be the point where ''DE'' crosses ''AB''. Then by the theorem, the equation also holds for ''D'', ''E'' and ''F''′. Comparing the two,
			: <math>\frac{AF}{FB} = \frac{AF'}{F'B}.</math>
			But at most one point can cut a segment in a given ratio so ''F''=''F''′.

			=== A non-computational proof using homothecies ===
			The following proof<ref>See Michèle Audin, Géométrie, éditions BELIN, Paris 1998: indication for exercice 1.37, p. 273</ref> uses only notions of [[affine geometry]], notably [[Homothetic transformation\|homothecies]].
			Whether or not ''D'', ''E'', ''F'' are collinear, there are three homothecies with centers ''D'', ''E'', ''F'' that respectively send ''B'' to ''C'', ''C'' to ''A'', and ''A'' to ''B''. The composition of the three then is an element of the group of homothecy-translations that fixes ''B'', so it is a homothecy with center ''B'', possibly with ratio 1 (in which case it is the identity). This composition fixes the line ''DE'' if and only if ''F'' is collinear with ''D'' and ''E'' (since the first two homothecies certainly fix ''DE'', and the third does so only if ''F'' lies on ''DE''). Therefore ''D'', ''E'', ''F'' are collinear if and only if this composition is the identity, which means that the product of the three ratios is 1:
			: <math>\frac{\overrightarrow{DC}}{\overrightarrow{DB}} \times
			\frac{\overrightarrow{EA}}{\overrightarrow{EC}} \times
			\frac{\overrightarrow{FB}}{\overrightarrow{FA}} = 1,</math>
			which is equivalent to the given equation.

			==History==
			It is uncertain who actually discovered the theorem; however, the oldest extant exposition appears in ''Spherics'' by Menelaus. In this book, the plane version of the theorem is used as a lemma to prove a spherical version of the theorem.<ref>{{cite book\|
			last=Smith\|first=D.E.\|title=History of Mathematics\|isbn=0-486-20430-8
			\|publisher=Courier Dover Publications\|year=1958\|volume=II\|page=607}}</ref>

			In [[Almagest]], [[Ptolemy]] applies the theorem on a number of problems in spherical astronomy.<ref name="rashed">{{cite book\|last=Rashed\|first=Roshdi\|title=Encyclopedia of the history of Arabic science\|volume=2\|page=483\|year=1996\|publisher=Routledge\|location=London\|isbn=0-415-02063-8}}</ref> During the [[Islamic Golden Age]], Muslim scholars devoted a number of works that engaged in the study of Menelaus' theorem, which they referred to as "the proposition on the secants" (''shakl al-qatta'''). The [[complete quadrilateral]] was called the "figure of secants" in their terminology.<ref name="rashed" /> [[Al-Biruni]]'s work, ''The Keys of Astronomy'', lists a number of those works, which can be classified into studies as part of commentaries on Ptolemy's ''Almagest'' as in the works of [[al-Nayrizi]] and [[al-Khazin]] where each demonstrated particular cases of Menelaus' theorem that led to the [[sine rule]],<ref name="musa">{{cite journal\|last=Moussa\|first=Ali\|title=Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations\|journal=Arabic Sciences and Philosophy\|year=2011\|volume=21\|issue=1\|publisher=[[Cambridge University Press]]\|doi=10.1017/S095742391000007X}}</ref> or works composed as independent treatises such as:

			* The "Treatise on the Figure of Secants" (''Risala fi shakl al-qatta''') by [[Thabit ibn Qurra]].<ref name="rashed" />
			* [[Husam al-DIn al-Salar]]'s ''Removing the Veil from the Mysteries of the Figure of Secants'' (Kashf al-qina' 'an asrar al-shakl al-qatta'), also known as "The Book on the Figure of Secants" (''Kitab al-shakl al-qatta''') or in Europe as ''The Treatise on the Complete Quadrilateral''. The lost treatise was referred to by [[Al-Tusi]] and [[Nasir al-Din al-Tusi]].<ref name="rashed" />
			* Work by [[al-Sijzi]].<ref name="musa" />
			* ''Tahdhib'' by [[Abu Nasr ibn Iraq]].<ref name="musa" />

			==References==
			{{Reflist}}
			* {{cite book \|title=Pure Geometry
			\|first=John Wellesley\|last=Russell\|publisher=Clarendon Press\|year=1905
			\|chapter= Ch. 1 §6 "Menelaus' Theorem"
			\|url=http://books.google.com/books?id=r3ILAAAAYAAJ}}

			== External links ==
			* [http://planetmath.org/?op=getobj&from=objects&id=3092 Alternate proof] of Menelaus' theorem, from [[PlanetMath]]
			* [http://www.cut-the-knot.org/Curriculum/Geometry/MenelausFromCeva.shtml Menelaus From Ceva]
			* [http://www.cut-the-knot.org/4travelers/CevaAndMenelaus.shtml Ceva and Menelaus Meet on the Roads]
			* [http://www.mathpages.com/home/kmath442/kmath442.htm Menelaus and Ceva] at MathPages
			* [http://demonstrations.wolfram.com/MenelausTheorem/ Menelaus' Theorem] by Jay Warendorff. [[The Wolfram Demonstrations Project]].
			* {{MathWorld \|title=Menelaus' Theorem \|urlname=MenelausTheorem}}

			[[Category:Affine geometry]]
			[[Category:Triangle geometry]]
			[[Category:Articles containing proofs]]
			[[Category:Theorems in plane geometry]]

en>Glrx: WP:ORDER; multi col by width

2012-04-29T03:22:54Z

WP:ORDER; multi col by width

New page

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