en>Jdaloner: /* Properties */ Linked "union" and "intersection" to "Union (set theory)" and "Intersection (set theory)" instead of disambiguation pages

2013-09-21T08:19:12Z

Properties: Linked "union" and "intersection" to "Union (set theory)" and "Intersection (set theory)" instead of disambiguation pages

← Older revision		Revision as of 10:19, 21 September 2013
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	Ed is ~~what individuals call me and my wife doesn~~'~~t like it at all~~. For ~~years she~~'s ~~been living in Kentucky but her husband desires them to move. My working day occupation~~ is ~~an invoicing officer but I've currently utilized for another~~ 1~~. What me~~ and ~~my family members adore~~ is ~~doing ballet but I've been using on new issues recently.~~<br><br>~~Also visit my~~ page; [http://~~checkmates~~.co.za/~~index~~.~~php?do=~~/~~profile~~-~~56347~~/~~info~~/ ~~online psychic~~] [http://~~www~~.~~octionx~~.~~sinfauganda~~.co.ug/~~node~~/~~22469 best psychics~~] ([http://~~fashionlinked~~.com/~~index~~.~~php?do~~=/~~profile-13453~~/~~info~~/ ~~fashionlinked~~.~~com~~])		[[File:Rouths theorem 1.png\|right\|260px]]

			In [[geometry]], '''Routh's theorem''' determines the ratio of areas between a given triangle and a triangle formed by the intersection of three [[cevian]]s. The theorem states that if in [[triangle]] <math>ABC</math> points <math>D</math>, <math>E</math>, and <math>F</math> lie on segments <math>BC</math>, <math>CA</math>, and <math>AB</math>, then writing <math>\tfrac{CD}{BD} </math><math>= x</math>, <math>\tfrac{AE}{CE} </math><math>= y</math>, and <math>\tfrac{BF}{AF} </math><math>= z</math>, the signed [[area]] of the triangle formed by the cevians <math>AD</math>, <math>BE</math>, and <math>CF</math> is the area of triangle <math>ABC</math> times

			: <math>\frac{(xyz - 1)^2}{(xy + y + 1)(yz + z + 1)(zx + x + 1)}.</math>

			This theorem was given by [[Edward John Routh]] on page 82 of his ''Treatise on Analytical Statics with Numerous Examples'' in 1896. The particular case <math> x = y = z = 2</math> has become popularized as the [[one-seventh area triangle]]. The <math> x = y = z = 1</math> case implies that the three [[median (geometry)\|median]]s are concurrent (through the [[centroid]]).

			==Proof==
			[[Image:Rouths theorem.png\|right\|250px]]

			Suppose the area of triangle<math>ABC</math> is 1. For triangle<math>ABD</math> and line<math>FRC</math> using [[Menelaus's theorem]], We could obtain:
			:<math>\frac{AF}{FB} \times \frac{BC}{CD} \times \frac{DR}{RA} = 1</math>
			Then<math>\frac{DR}{RA} = \frac{BF}{FA} \frac{DC}{CB} = \frac{zx}{x+1}</math>
			So the area of triangle<math>ARC</math> is:
			:<math>S_{ARC} = \frac{AR}{AD} S_{ADC} = \frac{AR}{AD} \frac{DC}{BC} S_{ABC} = \frac{x}{zx+x+1}</math>
			Similarly, we could know: <math>S_{BPA} = \frac{y}{xy+y+1}</math> and <math>S_{CQB} = \frac{z}{yz+z+1}</math>
			Thus the area of triangle<math>PQR</math> is:
			:<math>\displaystyle S_{PQR} = S_{ABC} - S_{ARC} - S_{BPA} - S_{CQB} </math>
			:<math>= 1 - \frac{x}{zx+x+1} - \frac{y}{xy+y+1} - \frac{z}{yz+z+1} </math>
			: <math>=\frac{(xyz - 1)^2}{(xz + x + 1)(yx + y + 1)(zy + z + 1)}.</math>

			== Citation ==
			The citation commonly given for Routh's Theorem is Routh's ''Treatise on Analytical Statics with Numerous Examples'', Volume 1, Chap. IV, in the [http://archive.org/details/atreatiseonanal00routgoog second edition] of 1896
			[http://archive.org/stream/atreatiseonanal00routgoog#page/n98/mode/1up p. 82], possibly because that edition has been easier to hand. However, Routh gave the theorem already in the [http://archive.org/details/cu31924001080237 first edition] of 1891, Volume 1, Chap. IV, [http://archive.org/stream/cu31924001080237#page/n102/mode/1up p. 89]. Although there is a change in pagination between the editions, the wording of the relevant footnote remained the same.

			Routh concludes his extended footnote with a ''caveat'':

			: ''The author has not met with these expressions for the areas of two triangles that often occur. He has therefore placed them here in order that the argument in the text may be more easily understood.''

			Presumably Routh felt those circumstances had not changed in the five years between editions. On the other hand, the title of Routh's book had been used earlier by [[Isaac Todhunter]]; both had been coached by [[William Hopkins]].

			Problems in this spirit have a long history in [[recreational mathematics]] and mathematical [[pedagogy\|paedagogy]], perhaps one of the oldest instances of being the determination of the proportions of the fourteen regions of the [[Stomachion]] board. With Routh's [[Cambridge]] in mind, the ''[[one-seventh_area_triangle\|one-seventh-area triangle]]'', associated in some accounts with [[Richard Feynman]], shows up, for example, as Question 100, [http://archive.org/stream/euclidselements00unkngoog#page/n91/mode/1up p. 80], in ''Euclid's Elements of Geometry ([http://archive.org/details/euclidselements01pottgoog Fifth School Edition])'', by [[Robert Potts]] (1805--1885,) of Trinity College, published in 1859; compare also his Questions 98, 99, on the same page. Potts stood twenty-sixth Wrangler in 1832 and then, like Hopkins and Routh, coached at Cambridge. Pott's expository writings in geometry were recognized by a [http://babel.hathitrust.org/cgi/pt?id=mdp.39015065371117#page/ii/mode/1up medal] at the International Exhibition of 1862, as well as by an Hon. LL.D. from the [[College_of_William_%26_Mary\|College of William and Mary]], [[Williamsberg]], [[Virginia]].

			== References ==
			* [[Murray S. Klamkin]] and A. Liu (1981) "Three more proofs of Routh's theorem", ''Crux Mathematicorum'' 7:199–203.
			*[[H. S. M. Coxeter]] (1969) ''Introduction to Geometry'', statement p. 211, proof pp. 219–20, 2nd edition, Wiley, New York.
			*J. S. Kline and D. Velleman (1995) "Yet another proof of Routh's theorem" (1995) ''Crux Mathematicorum'' 21:37–40
			* ''[http://demonstrations.wolfram.com/RouthsTheorem/ Routh's Theorem],'' Jay Warendorff, [[The Wolfram Demonstrations Project]].
			* {{MathWorld \|title=Routh's Theorem \|urlname=RouthsTheorem}}
			* [http://www.mathpages.com/home/kmath652/kmath652.htm Routh's Theorem by Cross Products] at MathPages
			* Ayoub, Ayoub B. (2011/2012) "Routh's theorem revisited", ''Mathematical Spectrum'' 44 (1): 24-27.

			[[Category:Triangle geometry]]
			[[Category:Area]]
			[[Category:Theorems in plane geometry]]
			[[Category:Affine geometry]]

en>Mike Rosoft: Reverted edits by 115.248.50.20 to last revision by Orange Suede Sofa (HG)

2010-11-14T09:25:48Z

Reverted edits by 115.248.50.20 to last revision by Orange Suede Sofa (HG)

New page

Ed is what individuals call me and my wife doesn't like it at all. For years she's been living in Kentucky but her husband desires them to move. My working day occupation is an invoicing officer but I've currently utilized for another 1. What me and my family members adore is doing ballet but I've been using on new issues recently.<br><br>Also visit my page; [http://checkmates.co.za/index.php?do=/profile-56347/info/ online psychic] [http://www.octionx.sinfauganda.co.ug/node/22469 best psychics] ([http://fashionlinked.com/index.php?do=/profile-13453/info/ fashionlinked.com])

Matrix grammar - Revision history

en>Jdaloner: /* Properties */ Linked "union" and "intersection" to "Union (set theory)" and "Intersection (set theory)" instead of disambiguation pages

en>Mike Rosoft: Reverted edits by 115.248.50.20 to last revision by Orange Suede Sofa (HG)