Example 4. As shown in the figure, points $A, B, C, D$ lie on the same circle, and $BC=DC=4, AE=6$. The lengths of segments $BE$ and $DE$ are both positive integers. What is the length of $BD$? (1988
National Junior High School Mathematics Competition)
Solve in $\triangle B C D$, $B C=D C$, H
Inference 1 gives
$$
C E^{2}=B C^{2}-B E \cdot D E,
$$
$\because A, B, C, D$ are concyclic,
$$
\begin{array}{l}
\therefore B E \cdot D E=A E \cdot C E, \\
\therefore C E^{2}=B C^{2}-A E \cdot C E=4^{2}-6 \times C E,
\end{array}
$$

which is $C E^{2}+6 \cdot C E-16=0$.
Solving gives $C E=2, C E=-8$ (discard),
$$
\begin{array}{l}
\therefore B E \cdot D E=A E \cdot C E=6 \times 2=12 . \\
\text { Also } B D<B C+C D=4+4=8, \\
\therefore\left\{\begin{array} { l } 
{ B E = 4 , } \\
{ D E = 3 ; }
\end{array} \text { or } \left\{\begin{array}{l}
B E=3, \\
D E=4 .
\end{array}\right.\right. \\
\therefore B D=7 .
\end{array}
$$