答案
A
解析
解：如图: 设 $B C=2 r_{1}, A B=2 r_{2}, A C=2 r_{3}$,
$$
\begin{aligned}
&\therefore r_{1}^{2}=r_{2}^{2}+r_{3}^{2}, \\
&\therefore S_{I}=\frac{1}{2} \times 4 r_{2} r_{3}=2 r_{2} r_{3}, S_{\text {III }}=\frac{1}{2} \times \pi r_{1}^{2}-2 r_{2} r_{3}, \\
&S_{\text {II }}=\frac{1}{2} \times \pi r_{3}^{2}+\frac{1}{2} \times \pi r_{2}^{2}-S_{\text {III }}=\frac{1}{2} \times \pi r_{3}^{2}+\frac{1}{2} \times \pi r_{2}{ }^{2}-\frac{1}{2} \times \pi r_{1}^{2}+2 r_{2} r_{3}=2 r_{2} r_{3}, \\
&\therefore S_{I}=S_{\text {II }}, \\
&\therefore P_{1}=P_{2},
\end{aligned}
$$
故选: A.