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 $area \, [MNPQ] \le \frac12 area\, [ABC]$.
The side of the square is $\frac{ah}{a+h}$ (can be verified by similar triangles easily enough ) where $a$ is the length of $BC$ and $h$ is the length of the height from $A$ to $BC$.
Note $h=\frac{2K}{a}$ where $K=[ABC]$.
So it suffices to show that \[\frac{4K^2}{a^2+4K^2/a^2+4K}\le \frac{K}{2}\]but expanding and rearranging shows that this is equivalent to
\[\left(a-\frac{2K}{a}\right)^2\ge 0\]evidently true.
\[\mathbb{Q.E.D.}\]