Example 1. In plane $a$, there is a regular $\triangle A^{\prime} B C$, and line $D E / / B C$ intersects $A^{\prime} B, A^{\prime} C$ at $D, E$ respectively. Now, $\triangle A^{\prime} D E$ is folded along $D E$ perpendicular to plane $\alpha$ to the position of $\triangle A D E$. When the line $D E$ parallel to $B C$ is taken at what position will $A B$ be the shortest?
(1979, Beijing Mathematical Competition)
Analyzing, we can set the length of the first line segment as $x$, then

Choosing, as shown in the figure, construct the altitude $A^{\prime} M$ of $\triangle A^{\prime} B C$, intersecting $D E$ at $N$. Since $D E / / B C$, we have $A^{\prime} N \perp D E$, i.e., $A N \perp D E$.
Also, since the plane $A D E \perp \alpha$, it follows that $A N \perp B N$.
Let the side length of the equilateral $\triangle A^{\prime} B C$ be $a, A^{\prime} N=A N=x$, then
$$
\begin{aligned}
A B^{2} & =A N^{2}+B N^{2} \\
& =x^{2}+\left(\frac{\sqrt{3}}{2} \cdot a-x\right)^{2}+\left(\frac{a}{2}\right)^{2} \\
& =2\left(x-\frac{\sqrt{3}}{4} a\right)^{2}+\frac{5}{8} a^{2} \geqslant \frac{5}{8} a^{2} .
\end{aligned}
$$