Minkowski's second theorem - Revision history

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	In [[~~mathematics~~]]~~, the '''perpendicular bisector construction''' is a construction which produces a new~~ [~~[quadrilateral]] from a given quadrilateral using the perpendicular bisectors to the sides of the former quadrilateral~~. ~~This construction arises very naturally and has been considered by many authors over the years~~.		My name is Penni and I am studying Architecture, Art, and Planning and Japanese Studies at Almere / Netherlands. <br>xunjie 瞬時には目を惹くシフォンレースレース、
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	~~==Definition of the construction==~~		Also visit my web site: [http://citruscontrols.com/Consulting/converse.html コンバースハイカット白]
	Suppose that the [[vertex (geometry)\|vertices]] of the [[quadrilateral]] <math> Q </math> are given by <math> Q_1,Q_2,Q_3,Q_4 </math>. Let <math> b_1,b_2,b_3,b_4 </math> be the perpendicular bisectors of sides <math> Q_1Q_2,Q_2Q_3,Q_3Q_4,Q_4Q_1 </math> respectively. Then their intersections <math> Q_i^{(2)}=b_{i+2}b_{i+3} </math>, with subscripts considered modulo 4, form the consequent quadrilateral <math> Q^{(2)} </math>. The construction is then iterated on <math> Q^{(2)} </math> to produce <math> Q^{(3)} </math> and so on.
	~~[[File:PerpendicularBisectorConstruction.svg\|thumb\|360px\|First iteration of the perpendicular bisector construction]]~~

	An equivalent construction can be obtained by letting the vertices of <math> Q^{(i+1)} </math> be the [[circumcenter]]s of the 4 triangles formed by selecting combinations of 3 vertices of <math> Q^{(i)} </math>.

	~~==Properties==~~
	1. If <math> Q^{(1)} </math> is not cyclic, then <math> Q^{(2)} </math> is not degenerate.<ref name=King>J. King, Quadrilaterals formed by perpendicular bisectors, in ''Geometry Turned On'', (ed. J. King), MAA Notes 41, 1997, pp. 29–32.</ref>

	~~2. Quadrilateral <math> Q^{(2)} </math> is never cyclic.<ref name="King" />~~

	3. Quadrilaterals <math> Q^{(1)} </math> and <math> Q^{(3)} </math> are [[Homothety\|homothetic]], and in particular, [[Similarity (geometry)\|similar]].<ref name=Shephard>G. C. Shephard, The perpendicular bisector construction, ''Geom. Dedicata'', 56 (1995) 75–84.</ref> Quadrilaterals <math> Q^{(2)} </math> and <math> Q^{(4)} </math> are also homothetic.

	3. The Perpendicular bisector construction can be reversed via [[isogonal conjugation]].<ref name=RadkoTsukerman>O. Radko and E. Tsukerman, The Perpendicular Bisector Construction, the Isoptic Point and the Simson Line of a Quadrilateral, ''Forum Geometricorum'' '''12''': ~~161–189 (2012).</ref> That is, given <math> Q^{(i+1)} </math>, it is possible to construct <math> Q^{(i)} </math>.~~

	4. Let <math> \alpha, \beta, \gamma, \delta </math> be the angles of <math> Q^{(1)} </math>. For every <math> i </math>, the ratio of areas of <math> Q^{(i)} </math> and <math> Q^{(i+1)} </math> is given by<ref name="RadkoTsukerman" />

	~~<div style="text-align: center;"><math> (1/4)(cot(\alpha)+cot(\gamma))(cot(\beta)+cot(\delta)). </math></div>~~

	5. If <math> Q^{(1)} </math> is convex then the sequence of quadrilaterals <math> Q^{(1)}, Q^{(2)},\ldots </math> converges to the [[Isoptic Point]] of <math> Q^{(1)} </math>, which is also the Isoptic Point for every <math> Q^{(i)} </math>. Similarly, if <math> Q^{(1)} </math> is concave, then the sequence <math> Q^{(1)}, Q^{(0)}, Q^{(-1)},\ldots </math> obtained by reversing the construction converges to the Isoptic Point of the <math> Q^{(i)} </math>'s.<ref name="RadkoTsukerman" />

	~~==Motivating discussion==~~
	~~The perpendicular bisector construction arises naturally in an attempt to find a replacement for the [~~[~~circumcenter]] of a quadrilateral in the case that it is noncyclic.~~

	~~==References==~~
	~~{{reflist}}~~
	<!--- After listing your sources please cite them using inline citations and place them after the information they cite. Please see http://en.wikipedia.org/wiki/Wikipedia:REFB for instructions on how to add citations. --->
	* J. Langr, Problem E1050, ''Amer. Math. Monthly'', 60 (1953) 551.
	* V. V. Prasolov, ''Plane Geometry Problems'', vol. 1 (in Russian), 1991; Problem 6.31.
	* V. V. Prasolov, ''Problems in Plane and Solid Geometry'', vol. 1 (translated by D. Leites), available at http://~~students~~.~~imsa.edu~~/~~~tliu~~/~~math/planegeo.eps.~~
	* D. Bennett, Dynamic geometry renews interest in an old problem, in ''Geometry Turned On'', (ed. J. King), MAA Notes 41, 1997, pp. 25–28.
	* J. King, Quadrilaterals formed by perpendicular bisectors, in ''Geometry Turned On'', (ed. J. King), MAA Notes 41, 1997, pp. 29–32.
	* G. C. ~~Shephard, The perpendicular bisector construction, ''Geom. Dedicata'', 56 (1995) 75–84.~~
	* A. Bogomolny, Quadrilaterals formed by perpendicular bisectors, ''Interactive Mathematics Miscellany and Puzzles'', http://www.cut-the-knot.org/Curriculum/Geometry/PerpBisectQuadri.shtml.
	* B. Grünbaum, On quadrangles derived from quadrangles—Part 3, ''Geombinatorics'' 7(1998), 88–94.
	* O. Radko and E. Tsukerman, The Perpendicular Bisector Construction, the Isoptic Point and the Simson Line of a Quadrilateral, ''Forum Geometricorum'' '''12''': 161–189 (2012).

	~~[[Category:Quadrilaterals]~~]

131.111.24.214: /* Statement of the theorem */ Replaced the Rs with \mathbb{R}s.

2013-08-30T14:02:10Z

Statement of the theorem: Replaced the Rs with \mathbb{R}s.

← Older revision		Revision as of 16:02, 30 August 2013
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Attempt~~ to ~~suit some form of these workout routines in your exercise.~~<br><br>~~Having various meats will allow you to construct your muscle tissues~~. ~~Create a objective of consuming 1 gram of lean beef for each lb of bodyweight that you are having~~. ~~This allows you to retail store greater amounts~~ of ~~health proteins and also~~ the ~~greater amount~~ of ~~proteins in your metabolism the greater your muscles can grow.~~<br><br>~~Range from~~ the ~~"large~~ 3~~" exercises with your exercise program~~. These bulk-creating workouts incorporate old lifts, squats and counter presses. These compound exercises could make you much stronger and enhance your muscle mass along with boosting your muscle mass. Make sure you attempt to feature a variance of most of these workout routines during your exercises routinely.<br><br>~~Locate an correct restriction for yourself~~, ~~but never give up until you reach the restrict you set~~. ~~For every established you do~~, ~~you must force oneself until finally you~~'~~re physically unable to lift up~~. ~~Try to force yourself to your boundaries~~. ~~If necessary~~, ~~minimize the time you spend on the packages once you get tired~~.<br><br>~~When using creatine monohydrate dietary supplements~~ is ~~beneficial~~, ~~exercising extreme care~~ in ~~the course of use~~, ~~particularly if making use of for extended time periods~~. ~~In case you have any type of trouble with your renal system, you should not take creatine~~. ~~In addition~~, ~~they are seen to result in muscle mass area disorder~~, ~~coronary [http://www~~.~~marktadvertenties~~.nl/~~author~~/~~hohajo~~/ ~~best weight muscle Gainer Supplement~~] ~~heart arrhythmia,~~ and ~~muscle mass cramping pains~~. ~~Creatine monohydrate ought to be particularly averted from~~ the ~~younger. Make certain you are just using~~ the ~~encouraged dosages with any health supplement and consult your doctor before you start use~~.<br><br>~~Possessing a smart routine focused on body building workout routines will allow you~~ to ~~develop muscles and keep you against injuries~~. ~~Individuals unfamiliar with routines about muscle building must reduce their routines to two times weekly~~. ~~More skillful men and women need to try~~ of ~~going about 3 x each week~~.<br><br>~~Think about plyometric workout routines. Plyometric exercises objective quickly-twitch muscles~~, ~~inspiring faster muscle progress. Plyometrics act like ballistic movements simply because that acceleration~~ ~~[http~~:/~~/sellsapps~~.~~com~~/~~author~~/~~jagrieve~~/ ~~sellsapps.com]~~ is ~~required. Squat jumps are a good example~~ of ~~plyometrics~~, ~~since as you may come up through the squat placement~~, ~~you jump up~~ to ~~feasible with your hands expanded towards~~ the ~~roof~~, ~~and~~ also ~~as you property, you enter in~~ the ~~squat before repeating the bounce~~.<br><br>~~When trying to acquire muscle~~, ~~try~~ to ~~eat foods rich in health proteins through~~ the ~~path~~ of the ~~time and just after your training session~~. ~~Strive~~ for ~~15 grams~~ of ~~health proteins both before~~ and after ~~you physical exercise~~. ~~This level of healthy proteins can be obtained from a glass or 2 of milk~~.~~<br~~>~~<br>If you are looking at bulking up~~, ~~you ought to give attention to counter pushing~~, ~~deceased lifting and squatting~~. ~~These 3 types of exercises can help you with getting into condition speedy~~ and ~~make muscle mass quickly~~. ~~When you have just about any inquiries relating to where~~ by ~~along with tips on how to employ [~~http://~~www~~.~~ad2post~~.~~com~~/~~author~~/~~mahudak~~/ ~~Best Weight Gainer Gnc]~~, ~~you possibly can call us from our web-page~~. ~~These a few are the major focuses~~, ~~but there can even be other [~~http://~~Deleukstefotos~~.~~nl/groups/what-everyone-should-know-about-getting-proper~~-~~nutrition/ exercises].<br><br>Perform some plyometric exercise routines. Incorporating these exercises to your regimen will improve~~ the ~~fast~~-~~twitch fibers inside your muscle groups and enhance muscle growth~~. ~~Plyometrics include acceleration to your [http:~~//~~911fashionpolice~~.~~com/index~~.~~php?do=/blog/31805/simple-ways-to-improve-your-diet-with-good-nutrition/ workout]~~. ~~As an example~~, ~~although undertaking plyometric drive-ups~~, ~~the hands need to jump up from the flooring~~, ~~exploding up to probable~~.~~<br><br>Begin using these pointers frequently if you exercise routine~~. ~~There are many benefits to muscle building, both both mental~~ and ~~physical~~. ~~Remain focused~~ and ~~specialized and you will probably see great results in your physical appearance~~, ~~and sense incredible brings about your strength and physical performing~~.		In [[mathematics]], the '''perpendicular bisector construction''' is a construction which produces a new [[quadrilateral]] from a given quadrilateral using the perpendicular bisectors to the sides of the former quadrilateral. This construction arises very naturally and has been considered by many authors over the years.

			==Definition of the construction==
			Suppose that the [[vertex (geometry)\|vertices]] of the [[quadrilateral]] <math> Q </math> are given by <math> Q_1,Q_2,Q_3,Q_4 </math>. Let <math> b_1,b_2,b_3,b_4 </math> be the perpendicular bisectors of sides <math> Q_1Q_2,Q_2Q_3,Q_3Q_4,Q_4Q_1 </math> respectively. Then their intersections <math> Q_i^{(2)}=b_{i+2}b_{i+3} </math>, with subscripts considered modulo 4, form the consequent quadrilateral <math> Q^{(2)} </math>. The construction is then iterated on <math> Q^{(2)} </math> to produce <math> Q^{(3)} </math> and so on.
			[[File:PerpendicularBisectorConstruction.svg\|thumb\|360px\|First iteration of the perpendicular bisector construction]]

			An equivalent construction can be obtained by letting the vertices of <math> Q^{(i+1)} </math> be the [[circumcenter]]s of the 4 triangles formed by selecting combinations of 3 vertices of <math> Q^{(i)} </math>.

			==Properties==
			1. If <math> Q^{(1)} </math> is not cyclic, then <math> Q^{(2)} </math> is not degenerate.<ref name=King>J. King, Quadrilaterals formed by perpendicular bisectors, in ''Geometry Turned On'', (ed. J. King), MAA Notes 41, 1997, pp. 29–32.</ref>

			2. Quadrilateral <math> Q^{(2)} </math> is never cyclic.<ref name="King" />

			3. Quadrilaterals <math> Q^{(1)} </math> and <math> Q^{(3)} </math> are [[Homothety\|homothetic]], and in particular, [[Similarity (geometry)\|similar]].<ref name=Shephard>G. C. Shephard, The perpendicular bisector construction, ''Geom. Dedicata'', 56 (1995) 75–84.</ref> Quadrilaterals <math> Q^{(2)} </math> and <math> Q^{(4)} </math> are also homothetic.

			3. The Perpendicular bisector construction can be reversed via [[isogonal conjugation]].<ref name=RadkoTsukerman>O. Radko and E. Tsukerman, The Perpendicular Bisector Construction, the Isoptic Point and the Simson Line of a Quadrilateral, ''Forum Geometricorum'' '''12''': 161–189 (2012).</ref> That is, given <math> Q^{(i+1)} </math>, it is possible to construct <math> Q^{(i)} </math>.

			4. Let <math> \alpha, \beta, \gamma, \delta </math> be the angles of <math> Q^{(1)} </math>. For every <math> i </math>, the ratio of areas of <math> Q^{(i)} </math> and <math> Q^{(i+1)} </math> is given by<ref name="RadkoTsukerman" />

			<div style="text-align: center;"><math> (1/4)(cot(\alpha)+cot(\gamma))(cot(\beta)+cot(\delta)). </math></div>

			5. If <math> Q^{(1)} </math> is convex then the sequence of quadrilaterals <math> Q^{(1)}, Q^{(2)},\ldots </math> converges to the [[Isoptic Point]] of <math> Q^{(1)} </math>, which is also the Isoptic Point for every <math> Q^{(i)} </math>. Similarly, if <math> Q^{(1)} </math> is concave, then the sequence <math> Q^{(1)}, Q^{(0)}, Q^{(-1)},\ldots </math> obtained by reversing the construction converges to the Isoptic Point of the <math> Q^{(i)} </math>'s.<ref name="RadkoTsukerman" />

			==Motivating discussion==
			The perpendicular bisector construction arises naturally in an attempt to find a replacement for the [[circumcenter]] of a quadrilateral in the case that it is noncyclic.

			==References==
			{{reflist}}
			<!--- After listing your sources please cite them using inline citations and place them after the information they cite. Please see http://en.wikipedia.org/wiki/Wikipedia:REFB for instructions on how to add citations. --->
			* J. Langr, Problem E1050, ''Amer. Math. Monthly'', 60 (1953) 551.
			* V. V. Prasolov, ''Plane Geometry Problems'', vol. 1 (in Russian), 1991; Problem 6.31.
			* V. V. Prasolov, ''Problems in Plane and Solid Geometry'', vol. 1 (translated by D. Leites), available at http://students.imsa.edu/~tliu/math/planegeo.eps.
			* D. Bennett, Dynamic geometry renews interest in an old problem, in ''Geometry Turned On'', (ed. J. King), MAA Notes 41, 1997, pp. 25–28.
			* J. King, Quadrilaterals formed by perpendicular bisectors, in ''Geometry Turned On'', (ed. J. King), MAA Notes 41, 1997, pp. 29–32.
			* G. C. Shephard, The perpendicular bisector construction, ''Geom. Dedicata'', 56 (1995) 75–84.
			* A. Bogomolny, Quadrilaterals formed by perpendicular bisectors, ''Interactive Mathematics Miscellany and Puzzles'', http://www.cut-the-knot.org/Curriculum/Geometry/PerpBisectQuadri.shtml.
			* B. Grünbaum, On quadrangles derived from quadrangles—Part 3, ''Geombinatorics'' 7(1998), 88–94.
			* O. Radko and E. Tsukerman, The Perpendicular Bisector Construction, the Isoptic Point and the Simson Line of a Quadrilateral, ''Forum Geometricorum'' '''12''': 161–189 (2012).

			[[Category:Quadrilaterals]]

Deltahedron: Deltahedron moved page User:Deltahedron/Minkowski's second theorem to Minkowski's second theorem: Publish

2012-07-03T18:17:56Z

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When using creatine monohydrate dietary supplements is beneficial, exercising extreme care in the course of use, particularly if making use of for extended time periods. In case you have any type of trouble with your renal system, you should not take creatine. In addition, they are seen to result in muscle mass area disorder, coronary [http://www.marktadvertenties.nl/author/hohajo/ best weight muscle Gainer Supplement] heart arrhythmia, and muscle mass cramping pains. Creatine monohydrate ought to be particularly averted from the younger. Make certain you are just using the encouraged dosages with any health supplement and consult your doctor before you start use. Possessing a smart routine focused on body building workout routines will allow you to develop muscles and keep you against injuries. Individuals unfamiliar with routines about muscle building must reduce their routines to two times weekly. More skillful men and women need to try of going about 3 x each week. 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