Example 5 As shown in Figure 2.10.5, let the perimeter of $\triangle A B C$ be $2 p$, and construct a tangent line $D E$ parallel to side $A C$ of the incircle of the triangle. Find the maximum length of the segment cut off by this tangent line on the other two sides.
Let $x$ be the length of the desired segment, $AC=b$, then the perimeter of $\triangle B E D$ equals $2 p-2 b$. From $\triangle B D E \sim \triangle B A C$, we have
$$
\frac{x}{b}=\frac{2 p-2 b}{2 p}
$$

which means
$$
x=\frac{1}{p} \cdot b(p-b)=\frac{1}{p}\left[\frac{p^{2}}{4}-\left(b-\frac{p}{2}\right)^{2}\right]
$$

Therefore, when $b=\frac{p}{2}$, $x$ reaches its maximum value $\frac{p}{4}$.