$20 \cdot 63$ A triangle has an inscribed circle with a radius of $r$ inches. If the perimeter of the triangle is $p$ inches and the area is $k$ square inches, then $\frac{p}{k}$ is
(A) independent of the value of $r$.
(B) $\frac{\sqrt{2}}{r}$.
(C) $\frac{2}{\sqrt{r}}$.
(D) $\frac{2}{r}$.
(E) $\frac{r}{2}$.
(18th American High School Mathematics Examination, 1967)
［Solution］From the figure, it is easy to see that
$$
\begin{aligned}
S_{\triangle A B C} & =\frac{1}{2} a r+\frac{1}{2} b r+\frac{1}{2} c r \\
& =\frac{1}{2}(a+b+c) r=\frac{1}{2} p r,
\end{aligned}
$$

Given $k=\frac{1}{2} p r$, then $\frac{p}{k}=\frac{2}{r}$. Therefore, the answer is $(D)$.