9. As shown in Figure 1, in $\triangle A B C$, $A B=9, B C=8$, $C A=7, \odot O_{1}$ passes through point $A$, and is tangent to line $B C$ at point $B, \odot O_{2}$ passes through point $A$, and is tangent to line $B C$ at point $C$. Let the other intersection point of $\odot O_{1}$ and $\odot O_{2}$, apart from point $A$, be $D$. Then $A D=$ $\qquad$
Figure 1
9. $\frac{33}{7}$.

Extend $A D$, intersecting $B C$ at point $E$.
By the power of a point theorem, we have $E B^{2}=E D \cdot E A=E C^{2}$.
Thus, $E$ is the midpoint of $B C$.
By the median length formula, we get
$$
\begin{array}{l}
E A^{2}=\frac{A B^{2}+A C^{2}}{2}-\frac{B C^{2}}{4}=49 \\
\Rightarrow E D=\frac{E B^{2}}{E A}=\frac{16}{7} \\
\Rightarrow A D=E A-E D=7-\frac{16}{7}=\frac{33}{7} \text {. } \\
\end{array}
$$