Example 7 As shown in the figure, $CD$ is the altitude on the hypotenuse of Rt $\triangle ABC$, $O$ is a point on $AC$, $OA$ $=OB=a$. Find: $OD^{2}+$ $CD^{2}=$ ?
Analysis: $C D^{2}=D A \cdot D B$, and the positions of $D A, D B, D O$ resemble the form of intersecting chords. Since $O A=O B$, we construct $\odot O$ with $O$ as the center and $O A$ as the radius, intersecting the extension of $D O$ at $E$ and $F$. We have
$$
\begin{array}{l}
D A \cdot D B=D E \cdot D F=(O E-O D)(O F+O D) \\
=a^{2}-O D^{2},
\end{array}
$$

Thus, $O D^{2}+C D^{2}=O D^{2}+D A \cdot D B=a^{2}$.