1. Points $A, B, C, D$ lie on the circumference of a circle, and $B C=C D=4, E$ is the intersection of $A C$ and $B D$, and $A E=6$. The lengths of segments $B E$ and $D E$ are both integers. What is the length of $B D$?
(1988 National Junior High School League Question)
1. From $\triangle C D E \sim \triangle B A E$ and $\triangle C B E \sim \triangle D A E$, we have $A B=\frac{4 B E}{C E}, A D=\frac{4 D E}{C E}$. Applying Ptolemy's theorem to quadrilateral $A B C D$, we get $B D \cdot(A E+C E)=4(A B+A D)=16 \cdot \frac{B E+D E}{C E}$. Let $C E=x$, we obtain the equation $x^{2}+6 x-16=0$, solving which gives $x=2$ (discarding the negative value). Thus, $B E \cdot D E=C E \cdot A E=12$. Also, $B D<B C+D C=8$, yielding $B E=3, D E=4$ or $B E=4, D E=3$, in either case $B D=7$ is the desired result.