Example 3 For a regular $n(n \geqslant 5)$-sided polygon centered at point $O$, let two adjacent points be denoted as $A, B, \triangle X Y Z$ is congruent to $\triangle O A B$. Initially, let $\triangle X Y Z$ coincide with $\triangle O A B$, then move $\triangle X Y Z$ on the plane so that points $Y$ and $Z$ both move along the perimeter of the polygon for one complete revolution, while point $X$ remains inside the polygon and moves accordingly. Find the trajectory of point $X$. (Refer to Example 10 in Chapter 10 of the first volume)
From
$$
\angle Y X Z+\angle Y B Z=\angle A O B+\angle A B C=\frac{n-2}{n} \pi+\frac{2}{n} \pi=\pi
$$

we have that points $X, Y, B, Z$ are concyclic, thus $\angle X B Z=\angle X Z Y=\angle O B Y$, which means $X$ lies on the extension of $B O$. Also,
$$
S_{\triangle X Y B}=\frac{1}{2} B X \cdot B Y \cdot \sin \angle X B Y=\frac{1}{2} X Y \cdot B Y \cdot \sin \angle X Y B
$$

we have
$$
B X=\frac{X Y \cdot \sin \angle X Y B}{\sin \angle X B Y}=\frac{X Y \cdot \sin \angle X Y B}{\sin \frac{n-2}{2 n} \pi}
$$
The maximum distance is
$$
d=\frac{X Y}{\sin \left(\frac{\pi}{2}-\frac{\pi}{n}\right)}-O B=\frac{a\left(1-\cos \frac{\pi}{n}\right)}{\sin ^{2} \frac{\pi}{n}}
$$

where $a$ is the side length of the regular $n$-gon. It is evident that as point $Y$ varies on $A B$, point $X$ precisely traces a segment of length $d$ on the extension of $B O$ from point $O$ twice. Thus, the locus of point $X$ is a "star" formed by segments of length $d$ extending from the center of the regular $n$-gon in the direction away from each vertex.