12 In an equilateral triangle $ABC$ with side length 1, take points $D$ and $E$ on sides $AB$ and $AC$ respectively, such that when the triangle is folded along segment $DE$, vertex $A$ lands exactly on side $BC$. The minimum length of $AD$ is $\qquad$.
12. Let $A D=x, \angle A D E=\alpha$, construct the symmetric figure of $\triangle A D E$ about $D E$, the symmetric point of $A$ is $G$ which falls on $B C$. In $\triangle D G B$,
$$
\frac{x}{\sin \frac{\pi}{3}}=\frac{1-x}{\sin \left(2 \alpha-\frac{\pi}{3}\right)} \Rightarrow x=\frac{\sqrt{3}}{\sqrt{3}+2 \sin \left(2 \alpha-\frac{\pi}{3}\right)} .
$$

When $\sin \left(2 \alpha-\frac{\pi}{3}\right)=1$, i.e., $x_{\text {min }}=\frac{\sqrt{3}}{2+\sqrt{3}}=2 \sqrt{3}-3$.