Inverting 7. Given a square with side length 1. Try to find the largest and the smallest area of an inscribed equilateral triangle within this square, and calculate these two areas (You must prove your argument).
(1978, National High School League)
Analysis: Let $\triangle E F G$ be
an inscribed equilateral triangle
in square $A B C D$. Since the three vertices of an
equilateral triangle must lie on at
least three sides of the square, we
can assume that points $F$ and $G$
lie on a pair of opposite sides of
the square.
Construct the altitude $E K$ of $\triangle E F G$, knowing that $E, K, G, D$ are concyclic $\Rightarrow \angle K D E=\angle K G E=60^{\circ}$. Similarly, $\angle K A E=60^{\circ}$. Therefore, $\triangle A D$ is also an equilateral triangle, and $K$ must be a fixed point.

The area of an equilateral triangle depends on its side length. When $K F \perp A B$, the side length is 1, which is the smallest, and the area $S=\frac{\sqrt{3}}{4}$ is also the smallest. When $K F$ passes through point $B$, the side length is $2 \cdot \sqrt{2-\sqrt{3}}$, which is the largest, and the area $S=2 \sqrt{3}-3$ is also the largest.