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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Euler's theorem in geometry ： ウィキペディア英語版 Euler's theorem in geometry In geometry, Euler's theorem states that the distance ''d'' between the circumcentre and incentre of a triangle can be expressed as〔.〕〔.〕〔.〕〔.〕 :$d^2=R\; (R-2r)\; \backslash ,$ or equivalently :$\backslash frac\; +\; \backslash frac\; =\; \backslash frac,$ where ''R'' and ''r'' denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively). The theorem is named for Leonhard Euler, who published it in 1767.〔.〕 However, the same result was published earlier by William Chapple in 1746.〔. The formula for the distance is near the bottom of p.123.〕 From the theorem follows the Euler inequality:〔〔 :$R\; \backslash ge\; 2r,$ which holds with equality only in the equilateral case.〔 ==Proof== Letting ''O'' be the circumcentre of triangle ''ABC'', and ''I'' be its incentre, the extension of ''AI'' intersects the circumcircle at ''L''. Then ''L'' is the midpoint of arc ''BC''. Join ''LO'' and extend it so that it intersects the circumcircle at ''M''. From ''I'' construct a perpendicular to AB, and let D be its foot, so ''ID'' = ''r''. It is not difficult to prove that triangle ''ADI'' is similar to triangle ''MBL'', so ''ID'' / ''BL'' = ''AI'' / ''ML'', i.e. ''ID'' × ''ML'' = ''AI'' × ''BL''. Therefore 2''Rr'' = ''AI'' × ''BL''. Join ''BI''. Because : ∠ ''BIL'' = ∠ ''A'' / 2 + ∠ ''ABC'' / 2, : ∠ ''IBL'' = ∠ ''ABC'' / 2 + ∠ ''CBL'' = ∠ ''ABC'' / 2 + ∠ ''A'' / 2, we have ∠ ''BIL'' = ∠ ''IBL'', so ''BL'' = ''IL'', and ''AI'' × ''IL'' = 2''Rr''. Extend ''OI'' so that it intersects the circumcircle at ''P'' and ''Q''; then ''PI'' × ''QI'' = ''AI'' × ''IL'' = 2''Rr'', so (''R'' + ''d'')(''R'' − ''d'') = 2''Rr'', i.e. ''d''2 = ''R''(''R'' − 2''r''). 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「Euler's theorem in geometry」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.