Example 6 As shown in Figure 3, in the isosceles trapezoid $A B C D$, $A B / / D C, A B$ $=998, D C=1001, A D=$ 1999, point $P$ is on the line segment $A D$. The number of points $P$ that satisfy the condition $\angle B P C=90^{\circ}$ is ( ).
(A) 0
(B) 1
(C) 2
(D) an integer not less than 3
Solution: Clearly, the angle subtended by $BC$ at the midpoint $M$ of $AD$ is $90^{\circ}$;

Take a point $N$ on $AD$ such that $AN=998$, then $DN=$ 1001. Since $\triangle ABN$ and $\triangle DCN$ are both isosceles triangles, it is easy to see that the angle subtended by $BC$ at $N$ is also $90^{\circ}$.

The circle with $BC$ as its diameter can intersect $AD$ at most at two points, so the number of points $P$ is 2. Therefore, the answer is $(C)$.