I3.4 In Figure $1, A P, A B, P B, P D, A C$ and $B C$ are line segments and $D$ is a point on $A B$. If the length of $A B$ is $R$ times that of $A D, \angle A D P=\angle A C B$ and $S=\frac{P B}{P D}$, find the value of $S$.
Consider $\triangle A D P$ and $\triangle A B P$.
$\angle A D P=\angle A C B=\angle A P B$ (given, $\angle$ s in the same segment $A B$ )
$\angle D A P=\angle P A B \quad$ (Common)
$\angle A P D=\angle A B P \quad(\angle \mathrm{s}$ sum of $\triangle)$
$\therefore \triangle A D P \sim \triangle A P B \quad$ (equiangular)
Let $A D=k, A B=5 k, A P=y$
$$
\begin{array}{l}
\frac{P B}{P D}=\frac{A B}{A P}=\frac{A P}{A D} \\
\frac{P B}{P D}=\frac{5 k}{y}=\frac{y}{k} \\
\therefore\left(\frac{y}{k}\right)^{2}=5 \Rightarrow \frac{y}{k}=\sqrt{5} \\
\frac{P B}{P D}=\sqrt{5}
\end{array}
$$
(Ratio of sides, $\sim \Delta$ 's)