Inegalitate proprie cu radicali intr-un triunghi.
Posted: Fri Dec 26, 2008 4:10 pm
Sa se arate ca in orice triunghi \( ABC \) exista inegalitatea \( \sum\sqrt {\frac {b + c - a}{a}} + 2\sum\frac {a}{b + c}\ \ge \ 6. \)
Aplicand ineg \( M_g\ge M_h \)
\( LHS=\sum {\sqrt{\frac{b+c-a}{a}}+2\sum{\frac{a}{b+c}}\ge\sum {\frac{2}{\frac{a}{b+c-a}+1}}+2\sum{\frac{a}{b+c}}=2\sum{\frac{b+c-a}{b+c}}+2\sum{\frac{a}{b+c}}=6=RHS \)