翻訳と辞書 Words near each other ・ Japan–European Union relations ・ Japan–Kenya relations ・ Japan–Korea Agreement of April 1905 ・ Japan–Korea Agreement of August 1904 ・ Japanese tea ceremony ・ Japanese Tea Garden ・ Japanese Tea Garden (San Francisco, California) ・ Japanese Teaching Staff Examination ・ Japanese Tears ・ Japanese television drama ・ Japanese temperate rainforest ・ Japanese Tenth Area Army ・ Japanese Terrier ・ Japanese the Manga Way ・ Japanese theorem ・ Japanese theorem for cyclic polygons ・ Japanese theorem for cyclic quadrilaterals ・ Japanese Third Area Army ・ Japanese Thirteenth Area Army ・ Japanese thrush ・ Japanese tissue ・ Japanese tit ・ Japanese topeshark ・ Japanese torpedo boat Kotaka ・ Japanese torpedo boat Tomozuru ・ Japanese Touring Car Championship ・ Japanese trade secrets ・ Japanese trademark law ・ Japanese traditional dance ・ Japanese traditional dolls Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Japanese theorem for cyclic polygons ： ウィキペディア英語版 Japanese theorem for cyclic polygons In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant.〔Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929).〕 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. ==Proof== This theorem can be proven by first proving a special case: no matter how one triangulates a cyclic quadrilateral, the sum of inradii of triangles is constant. After proving the quadrilateral case, the general case of the cyclic polygon theorem is an immediate corollary. 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. The quadrilateral case follows from a simple extension of the Japanese theorem for cyclic quadrilaterals, which 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 cyclic 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. Another proof of the quadrilateral case is available due to Wilfred Reyes (2002). In the proof, both the Japanese theorem for cyclic quadrilaterals and the quadrilateral case of the cyclic polygon theorem are proven as a consequence of Thébault's problem III. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Japanese theorem for cyclic polygons」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.