Example 3. $A, B$ are two fixed points, try to find point $C$ on the known line $l$ ($A, B$ are on the same side of $l$) such that $A C^{2}+B C^{2}$ is minimized.
Take the midpoint $M$ of $A B$, and take any point $X$ on $l$, connect $M X$, then in $\triangle A X B$, $X M$ is a median. By the median length formula, we have
$$
\begin{array}{l}
4 X_{M^{2}}=2\left(A X^{2}+B X^{2}\right)-A B^{2}, \\
\text { hence } A X^{2}+B X^{2}=\frac{4 X M^{2}+A B^{2}}{2} .
\end{array}
$$

The right side of the equation, $A B$, is a constant, so $A X^{2}+B X^{2}$ reaches its minimum $A C^{2}+B C^{2}$ only when $M X \perp l$, i.e., when $M C \perp l$ (where $C$ is the foot of the perpendicular).
This is the method of analysis.