从
\[2\sqrt{Rr}+2\sqrt{r_1r}+r_1=R，\]
得
\[
2\sqrt{r}(\sqrt{R}+\sqrt{r_1}) =R-r_1 \]
即
\[\begin{align}
2\sqrt{r}&= \sqrt{R}-\sqrt{r_1} \cr
&= \left(1-\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}\right)\sqrt{R}.
\end{align}\]
最终：
\[R=\frac{4r}{\left(1-\sqrt{(\sqrt{2}-1)/(\sqrt{2}+1)}\right)^2}.
\]

锟斤拷锟洁辑时锟斤拷: 2021-05-04 01:42:34