4. Points $A, B, C, D$ lie on the circumference of a circle, and $B C=D C=4, A E=6$, the lengths of segments $B E$ and $D E$ are both integers, find the length of $B D$.


4. For $\triangle B C D$ and point $E$ on side $B D$, applying Stewart's Theorem or its Corollary 1, we have $C E^{2}=4^{2} \cdot \frac{B E}{B D}+4^{2} \cdot \frac{D E}{B D}-B E \cdot$ $D E=16 \cdot \frac{B E+D E}{B D}-B E \cdot D E=16-B E \cdot D E=16-A E \cdot C E=16-6 C E$. Solving this, we get $C E=2$ (negative value discarded). Thus, $B E \cdot D E=C E \cdot A E=12$, and $B D<B C+C D=8$, which means $B E=3, D E=4$ or $B E=4, D E=3$, hence $B D=7$.