$20 \cdot 9$ As shown in the figure, $AB$ is the diameter of a semicircle, $O$ is the center, $CD \perp AB$, $DE \perp OC$. If the lengths of $AD$, $BD$, and $CD$ are all rational numbers, then among the following three propositions:
甲: The length of $OE$ is a rational number;
乙: The length of $DE$ is a rational number;
丙: The lengths of all line segments marked with letters in the figure are rational numbers. 
Among them,
(A) Only 甲 is correct.
(B) Only 乙 is correct.
(C) Only 甲 and 乙 are correct.
(D) 甲, 乙, and 丙 are all correct.
(China Junior High School Mathematics League, 1987)
[Solution] Since the lengths of $A D, B D$, and $C D$ are all rational numbers,
$O A=O B=O C=\frac{A D+B D}{2}$ is a rational number.
Thus, $O D=O A-A D$ is a rational number.
Therefore, $O E=\frac{O D^{2}}{O C}, \quad D E=\frac{D C \cdot D O}{O C}$ are both rational numbers.
Also, $C E=O C-O E, A B=A D+D B$ are rational numbers.
Therefore, statements 甲, 乙, and 丙 are all correct.
Hence, the answer is $(D)$.