[ Quadrilaterals (extreme properties).]

Among all parallelograms of a given area, find the one for which the largest diagonal is minimal.
Answer: square. Let $a$ and $b$ be the lengths of the sides of the parallelogram, $\alpha$ be the acute angle between its sides, $S$ be the area, and $d$ be the largest diagonal. Then $d^{2}=a^{2}+b^{2}+2 a b \cos \alpha$ and $a b \sin \alpha=S$. Therefore, $d^{2} \geq a^{2}+b^{2}$ and $a b \geq S$, and equality is achieved only when $\alpha=90^{\circ}$. Furthermore, $a^{2}+b^{2} \geq 2 a b$, so $d^{2} \geq 2 S$, and equality is achieved only when $a=b$ and $\alpha=90^{\circ}$, i.e., when the parallelogram is a square.