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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 British flag theorem ： ウィキペディア英語版 British flag theorem In Euclidean geometry, the British flag theorem says that if a point ''P'' is chosen inside rectangle ''ABCD'' then the sum of the squared Euclidean distances from ''P'' to two opposite corners of the rectangle equals the sum to the other two opposite corners.〔. Lardner includes this theorem in what he calls "the most useful and remarkable theorems which may be inferred" from the results in Book II of Euclid's Elements.〕〔.〕〔.〕 As an equation: : $AP^+CP^=BP^+DP^.\backslash ,$ The theorem also applies to points outside the rectangle, and more generally to the distances from a point in Euclidean space to the corners of a rectangle embedded into the space.〔(Harvard-MIT Mathematics Tournament solutions ), Problem 28.〕 Even more generally, if the sums of squared distances from a point ''P'' to the two pairs of opposite corners of a parallelogram are compared, the two sums will not in general be equal, but the difference of the two sums will depend only on the shape of the parallelogram and not on the choice of ''P''.〔.〕 == Proof == Drop perpendicular lines from the point ''P'' to the sides of the rectangle, meeting sides ''AB'', ''BC'', ''CD'', and ''AD'' in points ''w'', ''x'', ''y'' and ''z'' respectively, as shown in the figure; these four points ''wxyz'' form the vertices of an orthodiagonal quadrilateral. By applying the Pythagorean theorem to the right triangle ''AwP'', and observing that ''wP'' = ''Az'', it follows that * $AP^\; =\; Aw^\; +\; wP^\; =\; Aw^\; +\; Az^$ and by a similar argument the squared lengths of the distances from ''P'' to the other three corners can be calculated as * $PC^\; =\; wB^\; +\; zD^,$ * $BP^\; =\; wB^\; +\; Az^,$ and * $PD^\; =\; zD^\; +\; Aw^.$ Therefore: : $AP^\; +\; PC^\; =\; (Aw^\; +\; Az^)\; +\; (wB^\; +\; zD^)\; =\; (wB^\; +\; Az^)\; +\; (zD^\; +\; Aw^)\; =\; BP^\; +\; PD^.\backslash ,$ 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「British flag theorem」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.