Gọi D là trung điểm BC, lần lượt kẻ BP và CQ song song MN
\(\Rightarrow AG=\frac{2}{3}AD\)
\(\left\{{}\begin{matrix}\widehat{QCD}=\widehat{PBD}\left(slt\right)\\BD=DC\left(gt\right)\\\widehat{CDQ}=\widehat{PDQ}\left(dd\right)\end{matrix}\right.\) \(\Rightarrow\Delta CDQ=\Delta BDP\Rightarrow DQ=DP\)
\(MG//BP\Rightarrow\frac{AB}{AM}=\frac{AP}{AG}=\frac{AD+DP}{AG}\)
\(GN//CQ\Rightarrow\frac{AC}{AN}=\frac{AQ}{AG}=\frac{AD-DQ}{AG}\)
\(\Rightarrow\frac{AB}{AM}+\frac{AC}{AN}=\frac{AD+DP+AD-DQ}{AG}=\frac{2AD}{AG}=\frac{2AD}{\frac{2}{3}AD}=3\)