$20 \cdot 119$ As shown in the figure, three semicircles with diameters $AB$, $AC$, and $CB$ are pairwise tangent. If $CD \perp AB$, then the ratio of the area of the shaded region to the area of the circle with radius $CD$ is  
(A) $1: 2$.  
(B) $1: 3$.  
(C) $\sqrt{3}: 7$  
(D) $1: 4$.  
(E) $\sqrt{2}: 6$  
(18th American High School Mathematics Examination, 1967)
[Solution] Let $A_{1}, A_{2}, A_{3}$ represent the areas of the semicircles with diameters $A B=d_{1}, A C=d_{2}, B C=d_{3}$, respectively, $S$ represent the area of the shaded region, and $A$ represent the area of the circle with radius $C D=r$, i.e., $A=\pi r^{2}$. Thus,
$$
\begin{aligned}
S=A_{1}-A_{2}-A_{3} & =\frac{\pi}{8}\left(d_{1}^{2}-d_{2}^{2}-d_{3}^{2}\right) \\
& =\frac{\pi}{8}\left[\left(d_{2}+d_{3}\right)^{2}-d_{2}^{2}-d_{3}^{2}\right] \\
& =\frac{\pi}{4} d_{2} d_{3} .
\end{aligned}
$$

In the right triangle $\triangle A D B$, $C D^{2}=A C \cdot C B$, i.e., $r^{2}=d_{2} d_{3}$, then
$$
A=\pi r^{2}=\pi d_{2} d_{3}=4 S .
$$

Therefore, the answer is $(D)$.