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Saudi Arabian Mathematical Competitions

Geometry Difficulty 4.6 AIME Prove it Saudi Arabia

Let $ABC$ be an acute triangle and let $MNPQ$ be a square inscribed in the triangle such that $M, N \in BC$ , $P \in AC$ , $Q \in AB$ . Prove that
$\operatorname{area}[MNPQ] \leq \frac{1}{2} \operatorname{area}[ABC]$

Solution

Denote by $x$ the length of sides of square $MNPQ$ , $a = BC$ , $h_a = AA'$ , where $AA' \perp BC$ . The triangle $AQP$ and $ABC$ are similar, hence we have $\frac{x}{a} = \frac{h_a - x}{h_a}$ . We get
$x = \frac{a h_a}{a + h_a} \leq \frac{a h_a}{2 \sqrt{a h_a}} = \frac{1}{2} \sqrt{a h_a} = \frac{1}{2} \sqrt{2 \operatorname{area}[ABC]}$
and the desired inequality follows. The equality holds if and only if $h_a = a$ .

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