Example 5 Given that the parabola $y=-x^{2}+2 x+8$ intersects the $x$-axis at points $B$ and $C$, and point $D$ bisects $BC$. If point $A$ is a moving point on the parabola above the $x$-axis, and $\angle BAC$ is acute, then the range of $A i$ is $\qquad$
(1986, National Junior High School Mathematics League)
Analysis: From “$\angle B A C$ is an acute angle”, we know that point $A$ is outside the circle with the fixed line segment $B C$ as its diameter. Since point $A$ is above the $x$-axis, we can determine the range of the moving point $A$, and thus determine the range of $A D$.

Solution: As shown in Figure 5, the vertex of the given parabola is $A_{0}(1,9)$, the axis of symmetry is $x=1$, and it intersects the $x$-axis at two points $B(-2,0)$ and $C(4, 0)$.

Construct circles $\odot D$ and $\odot E$ with diameters $B C$ and $D A$, respectively. Then both circles intersect the parabola at two points $P(1-2 \sqrt{2}, 1)$ and $Q(1+2 \sqrt{2}, 1)$.

It is known that when point $A$ is within the parabolic arc $\overparen{P A_{0} Q}$ (excluding the endpoints), $\angle B A C<90^{\circ}$, and we have $3=D P=D Q<A D \leqslant D A_{0}=9$, i.e., the range of $A D$ is $3<A D \leqslant 9$.
2.3 Associating with the Power of a Point Theorem to construct auxiliary circles