Example 1 As shown in Figure 1, in square $A B C D$, $E$ is a fixed point on $B C$, and $B E=10, E C=14, P$ is a moving point on $B D$. Then the minimum value of $P E+P C$ is $\qquad$
(2006, Zhejiang Province Junior High School Mathematics Competition)
Analysis: Considering the minimum value of the sum of two line segments, it is relatively easy to think of using the property "the shortest distance between two points is a straight line." This requires transforming one of the line segments to the other side. Therefore, by the axial symmetry of the square (as shown in Figure 1), we know that $A P = P C$, so $P E + P C = A P + P E$. Thus, when point $P$ is on $A E$, $P E + P C$ is minimized, and by the Pythagorean theorem, the minimum value is 26.