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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 Bretschneider's formula ： ウィキペディア英語版 Bretschneider's formula In geometry, Bretschneider's formula is the following expression for the area of a general convex quadrilateral: :$K\; =\; \backslash sqrt\; \backslash right)\}$ ::$=\; \backslash sqrt\; abcd\; (1\; +\; \backslash cos\; (\backslash alpha+\; \backslash gamma))\}\; .$ Here, ''a'', ''b'', ''c'', ''d'' are the sides of the quadrilateral, ''s'' is the semiperimeter, and $\backslash alpha\; \backslash ,$ and $\backslash gamma\; \backslash ,$ are two opposite angles. Bretschneider's formula works on any convex quadrilateral, whether it is cyclic or not. The German mathematician Carl Anton Bretschneider discovered the formula in 1842. The formula was also derived in the same year by the German mathematician Karl Georg Christian von Staudt. == Proof of Bretschneider's formula == Denote the area of the quadrilateral by ''K''. Then we have : &= \frac + \frac. \end Therefore :$4K^2\; =\; (ad)^2\; \backslash sin^2\; \backslash alpha\; +\; (bc)^2\; \backslash sin^2\; \backslash gamma\; +\; 2abcd\; \backslash sin\; \backslash alpha\; \backslash sin\; \backslash gamma.\; \backslash ,$ The Law of Cosines implies that :$a^2\; +\; d^2\; -2ad\; \backslash cos\; \backslash alpha\; =\; b^2\; +\; c^2\; -2bc\; \backslash cos\; \backslash gamma,\; \backslash ,$ because both sides equal the square of the length of the diagonal ''BD''. This can be rewritten as :$\backslash frac\; =\; (ad)^2\; \backslash cos^2\; \backslash alpha\; +(bc)^2\; \backslash cos^2\; \backslash gamma\; -2\; abcd\; \backslash cos\; \backslash alpha\; \backslash cos\; \backslash gamma.\; \backslash ,$ Adding this to the above formula for $4K^2$ yields : &= (ad + bc)^2 - 4abcd \cos^2 \left(\frac\right). \end Following the same steps as in Brahmagupta's formula, this can be written as :$16K^2\; =\; (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)\; -\; 16abcd\; \backslash cos^2\; \backslash left(\backslash frac\backslash right).$ Introducing the semiperimeter :$s\; =\; \backslash frac,$ the above becomes :$16K^2\; =\; 16(s-a)(s-b)(s-c)(s-d)\; -\; 16abcd\; \backslash cos^2\; \backslash left(\backslash frac\backslash right)$ and Bretschneider's formula follows. 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「Bretschneider's formula」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.