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| In [[geometry]], the '''Japanese theorem''' states that the centers of the [[incircle]]s of certain [[triangles]] inside a [[cyclic quadrilateral]] are vertices of a rectangle.
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| Triangulating an arbitrary concyclic quadrilateral by its diagonals yields four overlapping triangles (each diagonal creates two triangles). The centers of the incircles of those triangles form a rectangle.
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| [[Image:Japanese theorem 2.svg|center|600px|]]
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| Specifically, let <math> \square ABCD </math> be an arbitrary concyclic quadrilateral and let be <math>M_1,M_2,M_3,M_4</math> the incenters of the triangles <math> \triangle ABD, \triangle ABC, \triangle BCD, \triangle ACD </math>. Then the quadrilateral formed by <math>M_1,M_2,M_3,M_4</math> is a rectangle.
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| Note that this theorem is easily extended to prove the [[Japanese theorem for cyclic polygons]]. To prove the quadrilateral case, simply construct the parallelogram tangent to the corners of the constructed rectangle, with sides parallel to the diagonals of the quadrilateral. The construction shows that the parallelogram is a rhombus, which is equivalent to showing that the sums of the radii of the incircles tangent to each diagonal are equal.
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| The quadrilateral case immediately proves the general case by induction on the set of triangulating partitions of a general polygon.
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| ==See also==
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| *[[Carnot's theorem]]
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| *[[Sangaku]]
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| *[[Wasan]]
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| ==References==
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| *Mangho Ahuja, Wataru Uegaki, Kayo Matsushita: [http://wayback.archive.org/web/*/http://www.math-cs.cmsu.edu/~mjms/2006.2/mangho999.ps ''In Search of the Japanese Theorem'']
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| *[http://www.cut-the-knot.org/proofs/jap.shtml Japanese Theorem at Cut-the-Knot]
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| *[http://www.gogeometry.com/sangaku2.html Japanese theorem, interactive proof with animation]
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| *Wataru Uegaki: "Japanese Theoremの起源と歴史" (On the Origin and History of the Japanese Theorem) http://hdl.handle.net/10076/4917
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| [[Category:Euclidean plane geometry]]
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| [[Category:Theorems in geometry]]
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| [[Category:Japanese mathematics]]
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