Example 5. Let $O$ be the center of the base $\triangle ABC$ of a regular tetrahedron $P-ABC$. A moving plane through $O$ intersects the three lateral edges or their extensions of the tetrahedron at points $Q$, $R$, and $S$ respectively. Then the sum $\frac{1}{PQ}+\frac{1}{PR}+\frac{1}{PS}(\quad)$.
(A) has a maximum value but no minimum value
(B) has a minimum value but no maximum value
(C) has both a maximum value and a minimum value, and they are not equal
(D) is a constant independent of the position of plane $QRS$
Analysis: This problem can be guessed from the following plane geometry problem. As shown in the figure, $P A=$
$P B, O$ is the midpoint of $A B$, and $R Q$ passes through point $O$ and intersects the extensions of $P B$ and $P A$ at $R$ and $Q$, respectively. By equal areas, we have
$$
\begin{aligned}
S_{\triangle P R Q} & =\frac{1}{2} P Q \cdot P R \sin \alpha \\
& =S_{\triangle O P Q}+S_{\triangle O P R} \\
& =\frac{1}{2} d \cdot P Q+\frac{1}{2} d \cdot P R .
\end{aligned}
$$

Thus, $\frac{1}{P Q}+\frac{1}{P R}=\frac{\sin \alpha}{d}$ is a constant.
By making a similar inference, the answer to this problem should be (D).