Example 21 As shown in Figure 1.4.23, in isosceles $\triangle ABC$, $AB=AC, \angle A=120^{\circ}$, point $D$ is on side $BC$, and $BD=1, DC=2$. Find the length of $AD$.
From the given conditions, we can cut $\triangle A B D$ and move it to the position of $\triangle A C D^{\prime}$. From $\angle A C B=\angle A B C=\angle C A D^{\prime}, A D=D^{\prime} C$, we know that quadrilateral $D C D^{\prime} A$ is an isosceles trapezoid.
Construct $A M \perp C D$ at $M$, then we can find
$$
A M=M C \cdot \tan 30^{\circ}=\frac{\sqrt{3}}{2} \cdot D M=\frac{1}{2}
$$

In the right $\triangle A D M$, we can find
$$
A D=\sqrt{A M^{2}+D M^{2}}=1
$$