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| [[Image:PonceletPorism.gif|thumb|right|Illustration of Poncelet's porism for ''n'' = 3, a triangle that is inscribed in one circle and circumscribes another.]]
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| In [[geometry]], '''Poncelet's porism''' (sometimes referred to as '''Poncelet's closure theorem'''), named after French engineer and mathematician [[Jean-Victor Poncelet]], states the following: Let ''C'' and ''D'' be two plane [[conic]]s. If it is possible to find, for a given ''n'' > 2, one ''n''-sided [[polygon]] that is simultaneously inscribed in ''C'' and circumscribed around ''D'' (i.e., a [[bicentric polygon]]), then it is possible to find infinitely many of them.
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| Poncelet's [[porism]] can be understood in terms of an [[elliptic curve]].
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| ==Sketch of proof==
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| View ''C'' and ''D'' as projective curves in '''P'''<sup>2</sup>. For simplicity, assume that ''C'' and ''D'' meet transversely. Then by [[Bézout's theorem]], ''C'' ∩ ''D'' consists of 4 (complex) points. For ''d'' in ''D'', let ''ℓ''<sub>''d''</sub> be the tangent line to ''D'' at ''d''. Let ''X'' be the subvariety of ''C'' × ''D'' consisting of (''c'',''d'') such that ''ℓ''<sub>''d''</sub> passes through ''c''. Given ''c'', the number of ''d'' with (''c'',''d'') ∈ ''X'' is 1 if ''c'' ∈ ''C'' ∩ ''D'' and 2 otherwise. Thus the projection ''X'' → ''C'' ≃ '''P'''<sup>1</sup> presents ''X'' as a degree 2 cover ramified above 4 points, so ''X'' is an elliptic curve (once we fix a base point on ''X''). Let <math>\sigma</math> be the involution of ''X'' sending a general (''c'',''d'') to the other point (''c'',''d''′) with the same first coordinate. Any involution of an elliptic curve with a fixed point, when expressed in the group law, has the form ''x'' → ''p'' − ''x'' for some ''p'', so <math>\sigma</math> has this form. Similarly, the projection ''X'' → ''D'' is a degree 2 morphism ramified over the contact points on ''D'' of the four lines tangent to both ''C'' and ''D'', and the corresponding involution <math>\tau</math> has the form ''x'' → ''q'' − ''x'' for some ''q''. Thus the composition <math>\tau \sigma</math> is a translation on ''X''. If a power of <math>\tau \sigma</math> has a fixed point, that power must be the identity. Translated back into the language of ''C'' and ''D'', this means that if one point ''c'' ∈ ''C'' (equipped with a corresponding ''d'') gives rise to an orbit that closes up (i.e., gives an ''n''-gon), then so does every point. The degenerate cases in which ''C'' and ''D'' are not transverse follow from a limit argument.
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| == See also ==
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| * [[Hartshorne ellipse]]
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| * [[Steiner's porism]]
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| * [[Tangent lines to circles]]
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| ==References==
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| *[[Bos, H. J. M.]]; Kers, C.; Oort, F.; Raven, D. W. Poncelet's closure theorem. Expositiones Mathematicae '''5''' (1987), no. 4, 289–364.
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| == External links ==
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| * [http://sbseminar.wordpress.com/2007/07/16/poncelets-porism/ David Speyer on Poncelet's Porism]
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| *D. Fuchs, S. Tabachnikov, ''Mathematical Omnibus: Thirty Lectures on Classic Mathematics''
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| * [http://www.borcherds.co.uk/PonceletsPorism.html Java applet] by Michael Borcherds showing the cases ''n'' = 3, 4, 5, 6, 7, 8 (including the convex cases for ''n'' = 7, 8) made using [http://www.geogebra.org/webstart/ GeoGebra].
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| * [http://www.borcherds.co.uk/geogebra/PonceletsPorismEllipseParabolaOrder3.html Java applet] by Michael Borcherds showing Poncelet's Porism for a general Ellipse and a Parabola made using [http://www.geogebra.org/webstart/ GeoGebra].
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| * [http://www.borcherds.co.uk/geogebra/PonceletsPorismEllipsesOrder3.html Java applet] by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 3) made using [http://www.geogebra.org/webstart/ GeoGebra].
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| * [http://www.borcherds.co.uk/geogebra/PonceletsPorismTwoConicsOrder5.html Java applet] by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 5) made using [http://www.geogebra.org/webstart/ GeoGebra].
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| * [http://www.borcherds.co.uk/geogebra/PonceletsPorismTwoConicsOrder6.html Java applet] by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 6) made using [http://www.geogebra.org/webstart/ GeoGebra].
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| * [http://poncelet.math.nthu.edu.tw/disk3/cabrijava/poncelet3-exterior2.html Java applet] showing the exterior case for n = 3 at National Tsing Hua University.
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| * [http://mathworld.wolfram.com/PonceletsPorism.html Article on Poncelet's Porism] at Mathworld.
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| [[Category:Conic sections]]
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| [[Category:Elliptic curves]]
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