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$\frac{AM}{MA_1}=\frac{S_{AMB}}{S_{MBA_1}}=\frac{S_{AMC}}{S_{MCA_1}}=\frac{S_{AMB}+S_{AMC}}{S_{BMC}}$
Tương tự $\frac{BM}{MB_1}=\frac{S_{AMB}+S_{BMC}}{S_{AMC}}$
$\frac{CM}{MC_1}=\frac{S_{AMC}+S_{BMC}}{S_{AMB}}$
$\frac{AM}{MA_1}+\frac{BM}{MB_1}+\frac{CM}{MC_1}$
$=\frac{S_{AMB}+S_{AMC}}{S_{BMC}}+\frac{S_{AMC}+S_{BMC}}{S_{AMB}}+\frac{S_{AMC}+S_{BMC}}{S_{AMB}}$
$=(S_{AMB}+S_{BMC}+S_{AMC})(\frac{1}{S_{AMB}}+ \frac{1}{S_{BMC}}+ \frac{1}{S_{AMC}})-3$
\geq $9-3=6$

$\frac{AM}{MA_1}.\frac{BM}{MB_1}.\frac{CM}{MC_1}$
$=\frac{(S_{AMB}+S_{BMC})(S_{BMC}+S_{AMC})(S_{AMC}+S_{AMB})}{S_{AMB}.S_{BMC}.S_{AMC}}$
\geq$\frac{2\sqrt{S_{AMB}.S_{BMC}}.2\sqrt{S_{BMC}.S_{AMC}}.2\sqrt{S_{AMC}.S_{AMB}}}{S_{AMB}.S_{BMC}.S_{AMC}}$
$=\frac{8S_{AMB}.S_{BMC}.S_{AMC}}{S_{AMB}.S_{BMC}.S_{AMC}}=8$
 
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