Example 5. As shown in Figure 5, in $\triangle ABC$, $AB=2$, $AC=3$, I, II, and III respectively denote the squares constructed on $AB$, $BC$, and $CA$. What is the maximum value of the sum of the areas of the three shaded regions? (1988, National Junior High School League)

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Solve in $\triangle A D E$ and $\triangle A B C$ !! $\angle B A C + \angle D A E = 180^{\circ}$.
Also, $A C = A D, A B = A E$,
thus $S \triangle A D B = S \triangle A B C$.
Similarly, the areas of the other two shaded triangles are also equal to $S \triangle \triangle B C$.
$$
\begin{array}{r}
\therefore S \text { shaded } = 3 S \triangle \mathrm{ABC} = 3 \cdot \frac{1}{2} A B \cdot A C \\
\cdot \sin \angle B A C \leqslant 3 \cdot \frac{1}{2} \cdot 2 \cdot 3 \cdot \sin 90^{\circ} = 9 .
\end{array}
$$