翻訳と辞書 Words near each other ・ Newton's inequalities ・ Newton's Institute of Engineering Macherla ・ Newton's law ・ Newton's law of cooling ・ Newton's law of universal gravitation ・ Newton's laws of motion ・ Newton's metal ・ Newton's method ・ Newton's method in optimization ・ Newton's Nation ・ Newton's parakeet ・ Newton's reflector ・ Newton's rings ・ Newton's sunbird ・ Newton's theorem ・ Newton's theorem (quadrilateral) ・ Newton's theorem about ovals ・ Newton's theorem of revolving orbits ・ Newton, Alabama ・ Newton, Brecknockshire ・ Newton, Bridgend ・ Newton, Chambers & Company ・ Newton, Cheshire ・ Newton, Chester ・ Newton, Cornwall ・ Newton, Derbyshire ・ Newton, Edmonton ・ Newton, England ・ Newton, Fenland ・ Newton, Georgia 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 Newton's theorem (quadrilateral) ： ウィキペディア英語版 Newton's theorem (quadrilateral) In Euclidean geometry Newton's theorem states, that in any tangential quadrilateral other than a rhombus the center of its incircle lies on its Newton line. Let ''ABCD'' be a tangential quadrilateral with at most one pair of parallel sides. Furthermore let ''E'' and ''F'' the midpoints of its diagonals ''AC'' and ''BD'' and ''P'' be the center of its incircle. Given such a configuration the point P is located on the Newton line, that is line ''EF'' connecting the midpoints of the diagonals. A tangential quadrilateral with two pairs of parallel sides is a rhombus. In this case both midpoints and the center of the incircle coincide and by definition no Newton line exists. Newton's theorem can easily be derived from Anne's theorem considering that in tangential quadrilaterals the combined lengths of opposite sides are equal (''a'' + ''c'' = ''b'' + ''d''). Now according to Anne's theorem showing that the combined areas of opposite triangles ''PAD'' and ''PBC'' and the combined areas of triangles ''PAB'' and ''PCD'' are equal is sufficient to ensure that ''P'' lies on ''EF''. Let ''r'' be the radius of the incircle, then ''r'' is also the altitude of all four triangles. :$\backslash begin$ &A(\triangle PAB)+A(\triangle PCD) \\ =&\tfracra+\tfracrc=\tfracr(a+c) \\ =&\tfracr(b+d)=\tfracrc+\tfracrd \\ =&A(\triangle PBC)+A(\triangle PAD) \end ==References== *Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. MAA, 2010, ISBN 9780883853481, pp. 117–118 () 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「Newton's theorem (quadrilateral)」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.