Example 2 As shown in Figure 2.10.2, consider a square with side length 1. Try to find the largest and the smallest inscribed equilateral triangles within this square, and calculate their areas.
Consider $\triangle E F G$ as an inscribed equilateral triangle in a square, without loss of generality, let $F, G$ be on $A B, C D$ respectively, and $E$ on $A D$.

Draw the altitude $E K$ from $\triangle E F G$ to side $F G$, with $K$ as the foot of the perpendicular, then $E, K, G, D$ are concyclic. Connect $K D$, then $\angle K D E=\angle E G K=60^{\circ}$.

Connect $A K$, similarly $\angle K A E=60^{\circ}$, so $\triangle K D A$ is an equilateral triangle, from which it follows that $K$ is a fixed point.
To maximize or minimize the side $F G$ with $K$ as the midpoint, it is sufficient to make $F$ coincide with $B$ (or $G$ with $C$) or to make $F G \parallel B C$, at this time $F G=2 \sqrt{2-\sqrt{3}}$ or $F G=1$, and $S_{\triangle E F G}=2 \sqrt{3}-3$ or $\frac{\sqrt{3}}{4}$ are the maximum and minimum areas of the equilateral triangle, respectively.