Ta có: \(\frac{AD}{OD}=\frac{S\left(ABC\right)}{S\left(OBC\right)};\frac{BE}{OE}=\frac{S\left(BAC\right)}{S\left(OAC\right)};\frac{CF}{OF}=\frac{S\left(CBA\right)}{S\left(OBA\right)}\)
=> \(\frac{AD}{OD}+\frac{BE}{OE}+\frac{CF}{OF}=S\left(ABC\right)\left(\frac{1}{S\left(OBC\right)}+\frac{1}{S\left(OAC\right)}+\frac{1}{S\left(OAB\right)}\right)\)\(\ge S\left(ABC\right)\left(\frac{9}{S\left(OBC\right)+S\left(OAC\right)+S\left(OAB\right)}\right)=\frac{S\left(ABC\right).9}{S\left(ABC\right)}=9\)
=> \(\frac{AD}{OD}+\frac{BE}{OE}+\frac{CF}{OF}\ge9\)
=> \(\frac{AO+OD}{OD}+\frac{BO+OE}{OE}+\frac{CO+OF}{OF}\ge9\)
=> \(\frac{AO}{OD}+\frac{BO}{OE}+\frac{CO}{OF}\ge6\)
Dấu "=" xảy ra <=> \(S\left(OBC\right)=S\left(OAC\right)=S\left(OAB\right)\)
