几何の暴力美学:角圆

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Exercise1: 如图两个圆称为角圆,求大圆与小圆的周长比与角度的关系

第一问很简单,解方程就是了:

\[\begin{aligned}
\sin \frac{\theta }{2} &= \frac{{AH}}{{AO}} = \frac{{BI}}{{BO}} \\
\sin \frac{\theta }{2} &= \frac{r}{{AO}} = \frac{R}{{r + R + AO}} \\
AO &= r\csc \frac{\theta }{2} = \frac{{r(R + r)}}{{R - r}} \\
\frac{R}{r} &= \frac{{\csc (\theta /2) + 1}}{{\csc (\theta /2) - 1}} = {\tan ^2}\left( {\frac{{\theta + \pi }}{4}} \right) \\
\end{aligned}\]


Exercise2:如果这样子一个一个又一个的圆无限延伸下去,求所有圆的总面积和三角形面积的比值.

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