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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 Conway triangle notation ： ウィキペディア英語版 Conway triangle notation In geometry, the Conway triangle notation, named after John Horton Conway, allows trigonometric functions of a triangle to be managed algebraically. Given a reference triangle whose sides are ''a'', ''b'' and ''c'' and whose corresponding internal angles are ''A'', ''B'', and ''C'' then the Conway triangle notation is simply represented as follows: :$S\; =\; bc\; \backslash sin\; A\; =\; ac\; \backslash sin\; B\; =\; ab\; \backslash sin\; C\; \backslash ,$ where ''S'' = 2 × area of reference triangle and :$S\_\backslash varphi\; =\; S\; \backslash cot\; \backslash varphi\; .\; \backslash ,$ in particular :$S\_A\; =\; S\; \backslash cot\; A\; =\; bc\; \backslash cos\; A=\; \backslash frac\; \backslash ,$ :$S\_B\; =\; S\; \backslash cot\; B\; =\; ac\; \backslash cos\; B=\; \backslash frac\; \backslash ,$ :$S\_C\; =\; S\; \backslash cot\; C\; =\; ab\; \backslash cos\; C=\; \backslash frac\; \backslash ,$ :$S\_\backslash omega\; =\; S\; \backslash cot\; \backslash omega\; =\; \backslash frac\; \backslash ,$      where $\backslash omega\; \backslash ,$ is the Brocard angle. :$S\_\}\; =\; S\; \backslash cot\; \}\; =\; S\; \backslash frac\; \backslash ,$ :$S\_\; =\; \backslash frac\; \backslash quad\backslash quad\; S\_\}\; =\; S\_\backslash varphi\; +\; \backslash sqrt\; \backslash ,$    for values of   $\backslash varphi$  where   :$S\_\; =\; \backslash frac\; \backslash quad\backslash quad\; S\_\; =\; \backslash frac\; \backslash ,$ Hence: :$\backslash sin\; A\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; \backslash ,$ Some important identities: :$\backslash sum\_\backslash text\; S\_A\; =\; S\_A+S\_B+S\_C\; =\; S\_\backslash omega\; \backslash ,$ :$S^2\; =\; b^2c^2\; -\; S\_A^2\; =\; a^2c^2\; -\; S\_B^2\; =\; a^2b^2\; -\; S\_C^2\; \backslash ,$ :$S\_BS\_C\; =\; S^2\; -\; a^2S\_A\backslash quad\backslash quad\; S\_AS\_C\; =\; S^2\; -\; b^2S\_B\backslash quad\backslash quad\; S\_AS\_B\; =\; S^2\; -\; c^2S\_C\; \backslash ,$ :$S\_AS\_BS\_C\; =\; S^2(S\_\backslash omega-4R^2)\backslash quad\backslash quad\; S\_\backslash omega=s^2-r^2-4rR\; \backslash ,$ where ''R'' is the circumradius and ''abc'' = 2''SR'' and where ''r'' is the incenter,   $s=\; \backslash frac\; \backslash ,$   and   $a+b+c\; =\; \backslash frac\; \backslash ,$ Some useful trigonometric conversions: :$\backslash sin\; A\; \backslash sin\; B\; \backslash sin\; C\; =\; \backslash frac\; \backslash quad\backslash quad\; \backslash cos\; A\; \backslash cos\; B\; \backslash cos\; C\; =\; \backslash frac$ :$\backslash sum\_\backslash text\; \backslash sin\; A\; =\; \backslash frac\; =\; \backslash frac\; \backslash quad\backslash quad\; \backslash sum\_\backslash text\; \backslash cos\; A\; =\; \backslash frac\; \backslash quad\backslash quad\; \backslash sum\_\backslash text\; \backslash tan\; A\; =\; \backslash frac\; =\backslash tan\; A\; \backslash tan\; B\; \backslash tan\; C\; \backslash ,$ Some useful formulas: :$\backslash sum\_\backslash text\; a^2S\_A\; =\; a^2S\_A\; +\; b^2S\_B\; +\; c^2\; S\_C\; =\; 2S^2\; \backslash quad\backslash quad\; \backslash sum\_\backslash text\; a^4\; =\; 2(S\_\backslash omega^2-S^2)\; \backslash ,$ :$\backslash sum\_\backslash text\; S\_A^2\; =\; S\_\backslash omega^2\; -\; 2S^2\; \backslash quad\backslash quad\; \backslash sum\_\backslash text\; S\_BS\_C\; =\; S^2\; \backslash quad\backslash quad\; \backslash sum\_\backslash text\; b^2c^2\; =\; S\_\backslash omega^2\; +\; S^2\; \backslash ,$ Some examples using Conway triangle notation: Let ''D'' be the distance between two points P and Q whose trilinear coordinates are ''p''''a'' : ''p''''b'' : ''p''''c'' and ''q''''a'' : ''q''''b'' : ''q''''c''. Let ''K''''p'' = ''ap''''a'' + ''bp''''b'' + ''cp''''c'' and let ''K''''q'' = ''aq''''a'' + ''bq''''b'' + ''cq''''c''. Then ''D'' is given by the formula: :$D^2=\; \backslash sum\_\backslash text\; a^2S\_A\backslash left(\backslash frac\; -\; \backslash frac\; \backslash right)^2\; \backslash ,$ Using this formula it is possible to determine OH, the distance between the circumcenter and the orthocenter as follows: For the circumcenter ''p''''a'' = ''aS''''A'' and for the orthocenter ''q''''a'' = ''S''''B''''S''''C''/''a'' :$K\_p=\; \backslash sum\_\backslash text\; a^2S\_A\; =\; 2S^2\; \backslash quad\backslash quad\; K\_q=\; \backslash sum\_\backslash text\; S\_BS\_C\; =\; S^2\; \backslash ,$ Hence: :$$ \begin D^2 & - \frac \right)^2 \\ & \sum_\text a^4S_A^3 - \frac \sum_\text a^2S_A + \frac \sum_\text S_BS_C \\ & \sum_\text a^2S_A^2(S^2-S_BS_C) - 2(S_\omega-4R^2) + (S_\omega-4R^2) \\ & \sum_\text a^2S_A^2 - \frac \sum_\text a^2S_A - (S_\omega-4R^2) \\ & \sum_\text a^2(b^2c^2-S^2) - \frac (S_\omega-4R^2) -(S_\omega-4R^2) \\ & - \frac \sum_\text a^2 - \frac (S_\omega-4R^2) \\ & S_\omega - \frac S_\omega + 6R^2 \\ & . ==References== * 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「Conway triangle notation」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. 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