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Geometry Difficulty 4.6 AIME Prove it Saudi Arabia

Let ABCABC be an acute triangle and let MNPQMNPQ be a square inscribed in the triangle such that M,N∈BCM, N \in BC, P∈ACP \in AC, Q∈ABQ \in AB. Prove that
area⁡[MNPQ]≤12area⁡[ABC] \operatorname{area}[MNPQ] \leq \frac{1}{2} \operatorname{area}[ABC]

Solution

Figure 1
Denote by xx the length of sides of square MNPQMNPQ, a=BCa = BC, ha=AA′h_a = AA', where AA′⊥BCAA' \perp BC. The triangle AQPAQP and ABCABC are similar, hence we have xa=ha−xha\frac{x}{a} = \frac{h_a - x}{h_a}. We get
x=ahaa+ha≤aha2aha=12aha=122area⁡[ABC] 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 ha=ah_a = a.

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