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| [[Image:Schoch Line.svg|thumb|400px|right|The Schoch line (cyan) passes through the point ''A''<sub>1</sub>.]]
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| In [[geometry]], the '''Schoch line''' is a [[Line (geometry)|line]] defined from an [[arbelos]] and named by Peter Woo after Thomas Schoch, who had studied it in conjunction with the [[Schoch circles]].
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| == Construction ==
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| An [[arbelos]] is a shape bounded by three mutually-tangent semicircular arcs with collinear endpoints, with the two smaller arcs nested inside the larger one; let the endpoints of these three arcs be (in order along the line containing them) ''A'', ''B'', and ''C''. Let ''K''<sub>1</sub> and ''K''<sub>2</sub> be two more arcs, centered at ''A'' and ''C'', respectively, with radii ''AB'' and ''CB'', so that these two arcs are tangent at ''B''; let ''K''<sub>3</sub> be the largest of the three arcs of the arbelos. A circle, with the center ''A''<sub>1</sub>, is then created [[tangent]] to the arcs ''K''<sub>1</sub>,''K''<sub>2</sub>, and ''K''<sub>3</sub>. This circle is congruent with [[Archimedes' circles|Archimedes' twin circles]], making it an [[Archimedean circle]]; it is one of the [[Schoch circles]]. The Schoch line is [[perpendicular]] to the line ''AC'' and passes through the point ''A''<sub>1</sub>. It is also the location of the centers of [[infinity|infinitely many]] Archimedean circles, e.g. the [[Woo circles]].<ref name="dswy">{{citation
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| | last1 = Dodge | first1 = Clayton W.
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| | last2 = Schoch | first2 = Thomas
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| | last3 = Woo | first3 = Peter Y.
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| | last4 = Yiu | first4 = Paul
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| | doi = 10.2307/2690883
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| | issue = 3
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| | journal = Mathematics Magazine
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| | mr = 1706441
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| | pages = 202–213
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| | title = Those ubiquitous Archimedean circles
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| | url = http://www.retas.de/thomas/arbelos/Ubiquitous.pdf
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| | volume = 72
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| | year = 1999}}.</ref>
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| == Radius and center of ''A''<sub>1</sub> ==
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| If ''r'' = ''AB''/''AC'', and ''AC'' = 1, then the radius of A<sub>1</sub> is
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| :<math>\rho=\frac{1}{2}r\left(1-r\right)</math>
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| and the center is
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| :<math>\left(r\left(1-r\right)\sqrt{\left(1+r\right)\left(2-r\right)}~,~\frac{1}{2}r\left(1+3r-2r^2\right)\right).</math><ref name="dswy"/>
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| == References ==
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| {{reflist}}
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| ==Additional reading==
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| *{{citation
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| | last1 = Okumura | first1 = Hiroshi
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| | last2 = Watanabe | first2 = Masayuki
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| | journal = Forum Geometricorum
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| | mr = 2057752
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| | pages = 27–34
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| | title = The Archimedean circles of Schoch and Woo
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| | url = http://forumgeom.fau.edu/FG2004volume4/FG200404.pdf
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| | volume = 4
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| | year = 2004}}.
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| == External links ==
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| *{{citeweb|author=van Lamoen, Floor|title=Schoch Line." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein|url=http://mathworld.wolfram.com/SchochLine.html|accessdate=2008-04-11}}
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| [[Category:Arbelos]]
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My hobby is mainly Computer programming. Sounds boring? Not!
I try to learn Korean in my spare time.
Here is my web blog app monetization