20. Let $A B C D$ be a parallelogram, and $P$ moves on the ray $A D$. $P B$ intersects $A C$ at $O$. Try to find the minimum value of $S(P)=\triangle A O P+\triangle B O C$.


20. Let $B C=a, A C=l, O C=x, \angle D A C=\alpha$, then
$$
\begin{array}{l}
S(P)=\frac{x^{2}}{2} \cdot \frac{\sin \alpha \sin (\alpha+P)}{\sin P}+\frac{(l-x)^{2}}{2} \cdot \frac{\sin \alpha \sin (\alpha+P)}{\sin P} \\
=\frac{1}{2} \sin \alpha\left[x^{2}+(l-x)^{2}\right] \frac{\alpha}{x}=\frac{\alpha}{2} \sin \alpha\left(2 x+\frac{l^{2}}{x}-2 l\right) \\
\geqslant \frac{a}{2} \sin \alpha(2 \sqrt{2} l-2 l)=(\sqrt{2}-1) a l \sin \alpha . \\
\end{array}
$$

When and only when $2 x=\frac{l^{2}}{x}$, i.e., $x=\frac{l}{\sqrt{2}}$, $S(P)=(\sqrt{2}-1) l a \sin \alpha$ is the minimum value sought.