gọi $I=BN\cap CM$$\Rightarrow (\overrightarrow{BN},\overrightarrow{CM})=\widehat{MIN}=\widehat{BIC}=120^{0}$
ta có
$\widehat{NBC}=180^{0}-120^{0}-\widehat{MCB}=60^{0}-\widehat{MCB}=\widehat{MCA}$
xét $\Delta NBC và \Delta MCA$ có:
$\widehat{MAC}=\widehat{NCB}=60^{0}$
$AC=BC$
$\widehat{NBC}=\widehat{MCA}$ (cmt)
suy ra $\Delta NBC =\Delta MCA$ (g_c_g)
$\Rightarrow NC=MA$
từ đó suy ra $\overrightarrow{AN}=\frac{2}{3}\overrightarrow{AC}\Rightarrow k=\frac{2}{3}$