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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 Equal incircles theorem ： ウィキペディア英語版 Equal incircles theorem In geometry, the equal incircles theorem derives from a Japanese Sangaku, and pertains to the following construction: a series of rays are drawn from a given point to a given line such that the inscribed circles of the triangles formed by adjacent rays and the base line are equal. In the illustration the equal blue circles define the spacing between the rays, as described. The theorem states that the incircles of the triangles formed (starting from any given ray) by every other ray, every third ray, etc. and the base line are also equal. The case of every other ray is illustrated above by the green circles, which are all equal. From the fact that the theorem doesn't depend on the angle of the initial ray, it can be seen that the theorem properly belongs to analysis, rather than geometry, and must relate to a continuous scaling function which defines the spacing of the rays. In fact, this function is the hyperbolic sine. The theorem is a direct corollary of the following lemma: Suppose that the nth ray makes an angle $\backslash gamma\backslash ,\_n$ with the normal to the baseline. If $\backslash gamma\backslash ,\_n$ is parameterized according to the equation, $\backslash tan\; \backslash gamma\backslash ,\_n\; =\; \backslash sinh\backslash theta\backslash ,\_n$, then values of $\backslash theta\backslash ,\_n\; =\; a\; +\; nb$, where $\backslash !a$ and $\backslash !b$ are real constants, define a sequence of rays that satisfy the condition of equal incircles, and furthermore any sequence of rays satisfying the condition can be produced by suitable choice of the constants $\backslash !a$ and $\backslash !b$. == Proof of the lemma == In the diagram, lines PS and PT are adjacent rays making angles $\backslash gamma\backslash ,\_n$ and $\backslash gamma\backslash ,\_$ with line PR, which is perpendicular to the baseline, RST. Line QXOY is parallel to the baseline and passes through O, the center of the incircle of $\backslash triangle$ PST, which is tangent to the rays at W and Z. Also, line PQ has length $\backslash !h-r$, and line QR has length $\backslash !r$, the radius of the incircle. Then $\backslash triangle$ OWX is similar to $\backslash triangle$ PQX and $\backslash triangle$ OZY is similar to $\backslash triangle$ PQY, and from XY = XO + OY we get : $(h-r)\; (\; \backslash tan\; \backslash gamma\backslash ,\_\; -\; \backslash tan\; \backslash gamma\backslash ,\_n\; )\; =\; r\; (\; \backslash sec\; \backslash gamma\backslash ,\_n\; +\; \backslash sec\; \backslash gamma\backslash ,\_\; ).$ This relation on a set of angles, $\backslash$, expresses the condition of equal incircles. To prove the lemma, we set $\backslash tan\; \backslash gamma\backslash ,\_n\; =\; \backslash sinh\; (a+nb)$, which gives $\backslash sec\; \backslash gamma\backslash ,\_n\; =\; \backslash cosh\backslash ,\; (a+nb)$. Using $\backslash !a+(n+1)b\; =\; (a+nb)+b$, we apply the addition rules for $\backslash !\backslash sinh$ and $\backslash !\backslash cosh$, and verify that the equal incircles relation is satisfied by setting : $\backslash frac\; =\; \backslash tanh\backslash frac.$ This gives an expression for the parameter $\backslash !b$ in terms of the geometric measures, $\backslash !h$ and $\backslash !r$. With this definition of $\backslash !b$ we then obtain an expression for the radii, $\backslash !r\_N$, of the incircles formed by taking every ''N''th ray as the sides of the triangles : $\backslash frac\; =\; \backslash tanh\backslash frac.$ 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「Equal incircles 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.