Example 8 Take any point $C_{1}$ on the directrix of the parabola, such that $C_{1}$ is not on the axis of symmetry of the parabola, and draw a tangent line to the parabola through $C_{1}$. Let the points of tangency be $A_{1}, B_{1}$, and the line $A_{1} B_{1}$ intersects the directrix at $C_{2}$. Then, draw a tangent line to the parabola through $C_{2}$, and let the points of tangency be $A_{2}, B_{2}$, and the line $A_{2} B_{2}$ intersects the directrix at $C_{3}$. Continue this process to obtain a sequence of points $C_{1}, C_{2}, C_{3}$, and try to find the characteristics of this sequence.
Let the parabola be $y^{2}=2 p x$, then $F\left(\frac{p}{2}, 0\right), C_{1}\left(-\frac{p}{2}, y_{1}\right), C_{2}\left(-\frac{1}{2} p, y_{2}\right)$.
By Corollary 2 of Property 3, we know that $A_{1} B_{1}$ passes through the focus $F$, and $A_{1} B_{1} \perp C_{1} F$, i.e., $C_{2} F \perp C_{1} F$. Thus, $-\frac{y_{1}}{-\frac{p}{2}-\frac{p}{2}} \cdot \frac{y_{2}}{-\frac{p}{2}-\frac{p}{2}}=-1$, which means $y_{1} y_{2}=-p^{2}$.

Similarly, $A_{2} B_{2}$ passes through the focus $F$, and $A_{2} B_{2}$, i.e., $C_{3} F \perp C_{2} F$. Thus, the lines $C_{1} F$ and $C_{3} F$ both pass through the focus $F$ and are perpendicular to $C_{2} F$, so $C_{3} F$ coincides with $C_{1} F$, and thus $C_{3}$ coincides with $C_{1}$.
By analogy, we can conclude that the sequence of points $C_{1}, C_{2}, C_{3}, \cdots$ has the following properties:
(1) $C_{2 n+1}=C_{1}, C_{2 n}=C_{2}(n$ is a positive integer $)$; (2) $y_{1} y_{2}=-p^{2}$.