Example 3 As shown in Figure 1, in the right trapezoid $A B C D$, $A B=7, A D=$ $2, B C=3$. If point $P$ on side $A B$ makes the triangle with vertices $P, A, D$ similar to the triangle with vertices $P, B, C$, then the number of such points $P$ is ( ).
(A) 1
(B) 2
(C) 3
(D) 4
Solution: (1) If $\triangle A P D \backsim \triangle B P C$, then $\frac{A P}{7-A P}=\frac{2}{3}$. Solving this, we get $A P=\frac{14}{5}$.
(2) If $\triangle A P D \backsim \triangle B C P$, then $\frac{A P}{3}=\frac{2}{7-A P}$. Solving this, we get $A P=1$ or 6. In summary, there are 3 points $P$, so the answer is (C). Note: Point $P$ has at least 1 and at most 3. Generally, if in the problem $A D=a, B C=b, A B=c, C D=d$. Then
(1) When $c^{2}-4 a b0$ i.e., $d>a+b$, point $P$ has 3.