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- Japanese_theorem_for_cyclic_polygons abstract "In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant. Conversely, if the sum of inradii independent from the triangulation, then the polygon is cyclic. The Japanese theorem follows from the Carnot's theorem; it is a Sangaku problem.This theorem also follows from a simple extension of the Japanese theorem for cyclic quadrilaterals.That theorem shows that a rectangle is formed by the two pairs of incenters corresponding to the two possible triangulations of the quadrilateral. The steps of this theorem require nothing beyond basic constructive Euclidean geometry.With the additional construction of a parallelogram having sides parallel to the diagonals, and tangent to the corners of the rectangle of incenters, the quadrilateral case of the concyclic polygon theorem can be proved in a few steps. The equality of the sums of the radii of the two pairs is equivalent to the condition that the constructed parallelogram be a rhombus, and this is easily shown in the construction.Also, it's readily shown that the quadrilateral case suffices to prove the general case of the concyclic polygon theorem. The quadrilateral rule can be applied to quadrilateral components of a general partition of a cyclic polygon, and repeated application of the rule, which "flips" one diagonal, will generate all the possible partitions from any given partition, with each "flip" preserving the sum of the inradii. Hence the concyclic polygon theorem considered here can be regarded as a corollary of the extended cyclic quadrilateral theorem.".
- Japanese_theorem_for_cyclic_polygons thumbnail Japanese_theorem_green.svg?width=300.
- Japanese_theorem_for_cyclic_polygons wikiPageExternalLink 4917.
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- Japanese_theorem_for_cyclic_polygons wikiPageExternalLink JapaneseTheorem.html.
- Japanese_theorem_for_cyclic_polygons wikiPageExternalLink mangho999.ps.
- Japanese_theorem_for_cyclic_polygons wikiPageID "2584965".
- Japanese_theorem_for_cyclic_polygons wikiPageRevisionID "590662445".
- Japanese_theorem_for_cyclic_polygons hasPhotoCollection Japanese_theorem_for_cyclic_polygons.
- Japanese_theorem_for_cyclic_polygons subject Category:Euclidean_plane_geometry.
- Japanese_theorem_for_cyclic_polygons subject Category:Japanese_mathematics.
- Japanese_theorem_for_cyclic_polygons subject Category:Theorems_in_geometry.
- Japanese_theorem_for_cyclic_polygons type Abstraction100002137.
- Japanese_theorem_for_cyclic_polygons type Communication100033020.
- Japanese_theorem_for_cyclic_polygons type Message106598915.
- Japanese_theorem_for_cyclic_polygons type Proposition106750804.
- Japanese_theorem_for_cyclic_polygons type Statement106722453.
- Japanese_theorem_for_cyclic_polygons type Theorem106752293.
- Japanese_theorem_for_cyclic_polygons type TheoremsInGeometry.
- Japanese_theorem_for_cyclic_polygons comment "In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant. Conversely, if the sum of inradii independent from the triangulation, then the polygon is cyclic.".
- Japanese_theorem_for_cyclic_polygons label "Japanese theorem for cyclic polygons".
- Japanese_theorem_for_cyclic_polygons label "Japanischer Satz für in einen Kreis einbeschriebene Polygone".
- Japanese_theorem_for_cyclic_polygons label "Japanse stelling".
- Japanese_theorem_for_cyclic_polygons label "Théorème japonais".
- Japanese_theorem_for_cyclic_polygons label "Twierdzenie japońskie".
- Japanese_theorem_for_cyclic_polygons label "مبرهنة يابانية في مضلع دائري".
- Japanese_theorem_for_cyclic_polygons sameAs Japanischer_Satz_für_in_einen_Kreis_einbeschriebene_Polygone.
- Japanese_theorem_for_cyclic_polygons sameAs Théorème_japonais.
- Japanese_theorem_for_cyclic_polygons sameAs 일본인의_정리.
- Japanese_theorem_for_cyclic_polygons sameAs Japanse_stelling.
- Japanese_theorem_for_cyclic_polygons sameAs Twierdzenie_japońskie.
- Japanese_theorem_for_cyclic_polygons sameAs m.07ps1y.
- Japanese_theorem_for_cyclic_polygons sameAs Q1317367.
- Japanese_theorem_for_cyclic_polygons sameAs Q1317367.
- Japanese_theorem_for_cyclic_polygons sameAs Japanese_theorem_for_cyclic_polygons.
- Japanese_theorem_for_cyclic_polygons wasDerivedFrom Japanese_theorem_for_cyclic_polygons?oldid=590662445.
- Japanese_theorem_for_cyclic_polygons depiction Japanese_theorem_green.svg.
- Japanese_theorem_for_cyclic_polygons isPrimaryTopicOf Japanese_theorem_for_cyclic_polygons.