5. In square $A B C D$, $E$ is on $B C$, $B E=2, C E=1, P$ is a point on $B D$, then what is the minimum possible value of the sum of the lengths of $P E$ and $P C$?
5. Given $B E: C E=2$, for $\triangle B C P$ and point $E$ on side $B C$, applying Corollary 5 of Stewart's Theorem, we have $P E^{2}=\frac{1}{3} P B^{2}+ \frac{2}{3} P C^{2}-2$. For $\triangle B C D$ and point $P$ on side $B D$, applying Corollary 1, we have $P C^{2}=B C^{2}-B P \cdot P D=9+P B^{2}-3 \sqrt{2} P B$, thus $P E^{2}=P B^{2}-2 \sqrt{2} P B+4$, hence $P E+P C=\sqrt{(P B-\sqrt{2})^{2}+(0-\sqrt{2})^{2}}+\sqrt{\left(P B-\frac{3 \sqrt{2}}{2}\right)^{2}+\left(0+\frac{3 \sqrt{2}}{2}\right)^{2}}$. Let $P B=x$, the above expression represents the sum of the distances from a moving point $Q(x, 0)$ on the $x$-axis to two fixed points $A(\sqrt{2}, \sqrt{2}), B\left(\frac{3 \sqrt{2}}{2},-\frac{3 \sqrt{2}}{2}\right)$. When $Q$ is the intersection point of segment $A B$ with the $x$-axis $\left(\frac{6}{5} \sqrt{2}, 0\right)$, i.e., $P B=\frac{6}{5} \sqrt{2}$, $P E+P C$ achieves its minimum value $\sqrt{13}$.