$20 \cdot 112$ As shown in the figure, the diameter of circle $K$ is $AB$, circle $L$ is tangent to circle $K$ and tangent to diameter $AB$ at the center of circle $K$; circle $M$ is tangent to circle $K$, circle $L$, and diameter $AB$. The ratio of the area of circle $K$ to the area of circle $M$ is
(A) 12.
(B) 14.
(C) 16.
(D) 18.
(E) Not an integer.
(27th American High School Mathematics Examination, 1976)
[Solution] Connect $L K, L M, K M$, and draw $M F$ parallel to $A B$ 
intersecting $K L$ at $F$. Let $r, s(=r / 2)$, and $t$ be the radii of the circles with centers at $K, L$, and $M$ respectively. In $\triangle F L M$ and $\triangle F K M$, by the Pythagorean theorem, we have
$$
\begin{array}{l}
M F^{2}=\left(\frac{r}{2}+t\right)^{2}-\left(\frac{r}{2}-t\right)^{2}, \\
M F^{2}=(r-t)^{2}-t^{2} . \\
\text { From }\left(\frac{r}{2}+t\right)^{2}-\left(\frac{r}{2}-t\right)^{2}=(r-t)^{2}-t^{2} \text {, } \\
\end{array}
$$

we get $\frac{r}{t}=4$. Therefore, the required ratio is 16.
Hence, the answer is $(C)$.