G5.4 In figure $3, \triangle A B C$ is a right-angled triangle and $B H \perp A C$. If $A B=15, H C=16$ and the area of $\triangle A B C$ is $S$, find the value of $S$.
It is easy to show that $\triangle A B H \sim \triangle B C H \sim \triangle A C B$.
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\begin{array}{l}
\text { Let } \angle A B H=\theta=\angle B C H \\
\text { In } \triangle A B H, B H=15 \cos \theta \\
\text { In } \triangle B C H, C H=B H \div \tan \theta \Rightarrow 16 \tan \theta=15 \cos \theta \\
16 \sin \theta=15 \cos ^{2} \theta \Rightarrow 16 \sin \theta=15-15 \sin ^{2} \theta \\
15 \sin ^{2} \theta+16 \sin \theta-15=0 \\
(3 \sin \theta+5)(5 \sin \theta-3)=0 \\
\sin \theta=\frac{3}{5} ; \tan \theta=\frac{3}{4} \\
B C=A B \div \tan \theta=15 \times \frac{4}{3}=20 \\
\text { Area of } \triangle A B C=\frac{1}{2} \cdot 15 \times 20=150=S
\end{array}
$$