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| In [[plane geometry]], '''Holditch's theorem''' states that if a [[chord (geometry)|chord]] of fixed length is allowed to rotate inside a convex closed curve, then the [[locus (mathematics)|locus]] of a point on the chord a distance ''p'' from one end and a distance ''q'' from the other is a closed curve whose area is less than that of the original curve by <math>\pi pq</math>. The theorem was published in 1858 by Rev. [[Hamnet Holditch]].<ref name=Pickover/><ref>Holditch, Rev. Hamnet, "Geometrical theorem", ''[[The Quarterly Journal of Pure and Applied Mathematics]]'' 2, 1858, p. 38.</ref> While not mentioned by Holditch, the proof of the theorem requires an assumption that the chord be short enough that the traced locus is a simple closed curve.<ref name=Broman>[[Arne Broman|Broman, Arne]], [http://www.jstor.org/stable/2689793 "A fresh look at a long-forgotten theorem"], ''[[Mathematics Magazine]]'' 54(3), May 1981, 99–108.</ref>
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| ==Observations==
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| The theorem is included as one of [[Clifford Pickover]]'s 250 milestones in the history of mathematics.<ref name=Pickover>{{citation |last=Pickover|first=Clifford|title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics |publisher=Sterling |date=1 September 2009|page=250 |isbn = 978-1-4027-5796-9}}</ref> Some peculiarities of the theorem include that the area formula <math>\pi pq</math> is independent of both the shape and the size of the original curve, and that the area formula is the same as for that of the area of an [[ellipse]] with semi-axes ''p'' and ''q''. The theorem's author was a president of [[Caius College, Cambridge]].
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| ==Extensions==
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| Broman<ref name=Broman/> gives a more precise statement of the theorem, along with a generalization. The generalization allows, for example, consideration of the case in which the outer curve is a [[triangle]], so that the conditions of the precise statement of Holditch's theorem do not hold because the paths of the endpoints of the chord have [[retrograde motion|retrograde]] portions (portions that retrace themselves) whenever an [[acute angle]] is traversed. Nevertheless, the generalization shows that if the chord is shorter than any of the triangle's [[altitude (triangle)|altitude]]s, and is short enough that the traced locus is a simple curve, Holditch's formula for the in-between area is still correct (and remains so if the triangle is replaced by any [[convex polygon]] with a short enough chord). However, other cases result in different formulas.
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| ==References==
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| {{reflist}}
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| ==References not as yet cited in text==
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| * [http://www.nature.com/nature/journal/v96/n2411/abs/096541a0.html B. Williamson, FRS], An elementary treatise on the integral calculus : containing applications to plane curves and surfaces, with numerous examples (Longmans, Green, London, 1875; 2nd 1877; 3rd 1880; 4th 1884; 5th 1888; 6th 1891; 7th 1896; 8th 1906; 1912, 1916, 1918, 1926); [http://archive.org/details/anelementarytre02willgoog Ist 1875], pp. 192–193, with citation of Holditch's Prize Question set in The Lady's and Gentleman's Diary for 1857 (appearing in late 1856), with extension by Woolhouse in the issue for 1858; [http://archive.org/details/cu31924031264769 5th 1888]; [http://archive.org/details/elementarytreati00willuoft 8th 1906] pp. 206–211
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| * J. Edwards, [http://archive.org/details/treatiseonintegr01edwauoft A Treatise on the Integral Calculus with Applications, Examples and Problems, Vol. 1] (Macmillan, London, 1921), Chap. XV, esp. Sections 478, 481–491, 496 (see also Chap. XIX for instantaneous centers, roulettes and glisettes); expounds and references extensions due to Woolhouse, Elliott, Leudesdorf, Kempe, drawing on the earlier book of Williamson.
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| * E. Kilic and S. Keles, [http://dergiler.ankara.edu.tr/dergiler/29/1409/15929.pdf On Holditch's Theorem and Polar Inertia Momentum], Commun. Fac. Sci. Univ. Ank. Ser. A, 43 (1994), 41–47.
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| * M. J. Cooker, [http://www.jstor.org/stable/3620400 An Extension of Holditch's Theorem on the Area within a Closed Curve], Math. Gaz., 82 (1998), 183–188.
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| * M. J. Cooker, [http://www.jstor.org/stable/3618685 On Sweeping out an Area],Math. Gaz., 83 (1999), 69–73.
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| * T. M. Apostol, with Mamikon A. Mnatsakanian, [http://books.google.com/books?id=PUVvwfjhjvMC New Horizons in Geometry. Dolciani Mathematical Expositions 47] (Math. Assoc. Amer., Washington, DC, 2013), Section 9.13
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| ==External links==
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| * [http://mathworld.wolfram.com/HolditchsTheorem.html MathWorld article]
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| {{DEFAULTSORT:Holditch's Theorem}}
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| [[Category:Theorems in geometry]]
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