Problem

Source: 2010 Saudi Arabia BMO TST 1.2 - Balkan

Tags: geometry, areas, geometric inequality, inscribed, square



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 $area \, [MNPQ] \le \frac12 area\, [ABC]$.