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nhuquynhdat

Ta có: $\dfrac{AM}{OM}=\dfrac{S_{AMC}}{S_{OMC}}=\dfrac{S_{AMB}}{S_{BMO}}=\dfrac{S_{ABC}}{S_{BOC}}$

$\Longrightarrow \dfrac{AM}{OM}=\dfrac{S_{ABC}}{S_{BOC}}$

Hoàn toàn tương tự, ta có:

$\dfrac{BN}{ON}=\dfrac{S_{ABC}}{S_{AOC}}$

$\dfrac{CP}{OP}=\dfrac{S_{ABC}}{S_{AOB}}$

$\Longrightarrow \dfrac{AM}{OM}+ \dfrac{BN}{ON}+ \dfrac{BN}{ON}= \dfrac{S_{ABC}}{S_{BOC}}+ \dfrac{S_{ABC}}{S_{AOC}}+ \dfrac{S_{ABC}}{S_{AOB}}= S_{ABC}.(\dfrac{1}{S_{AOB}}+ \dfrac{1}{S_{AOC}}+ \dfrac{1}{S_{BOC}})=(S_{AOC}+S_{AOB}+S_{BOC})( \dfrac{1}{S_{AOB}}+ \dfrac{1}{S_{AOC}}+ \dfrac{1}{S_{BOC}}) \ge 9$
 
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