Example 5. Find a point P on a fixed line L such that $\mathrm{PA}^{2} + PB^{2}$ is minimized (A and B are two fixed points).
Solve: As shown in Figure 6, take the midpoint C of AB, and let P be the point on L that satisfies the condition, then $PA^2 + PB^2 = 2(PC^2 + AC^2)$, so PC must be the shortest. Therefore, draw CP perpendicular to L through C, and the foot of the perpendicular P is the desired point.