Inegalitate in triunghi ascutitunghic

[Cezar Lupu]

Sa se arate ca in orice triunghi ascutitunghic de laturi \( a, b, c \) are loc inegalitatea

\( \sum_{cyc}\sqrt{(b^2+c^2-a^2)(c^2+a^2-b^2)}\leq\sum_{cyc}ab \).

Murray Klamkin, AMM

[Marius Mainea]

Folosim inegalitatea \( \sum {\cos A\cos B}\le \frac{3}{4} \) si CBS de unde

\( LHS=\sum {\sqrt{2bc\cos A2ca\cos B}}\le\sqrt{4abc(a+b+c)\sum {\cos A\cos B}}\le RHS \)