c)
\(\begin{array}{l}\overrightarrow {AI} = \dfrac{3}{5}\overrightarrow {AM} = \dfrac{3}{5}\left( {\dfrac{1}{3}\overrightarrow {AB} + \dfrac{2}{3}\overrightarrow {AC} } \right) = \dfrac{1}{5}\overrightarrow {AB} + \dfrac{2}{5}\overrightarrow {AC} \\\overrightarrow {AN} = \dfrac{1}{2}\overrightarrow {AC} \\ \Rightarrow \overrightarrow {BI} = \overrightarrow {AI} - \overrightarrow {AB} = \dfrac{1}{5}\overrightarrow {AB} + \dfrac{2}{5}\overrightarrow {AC} - \overrightarrow {AB} = - \dfrac{4}{5}\overrightarrow {AB} + \dfrac{2}{5}\overrightarrow {AC} \\\overrightarrow {BN} = \overrightarrow {BA} + \overrightarrow {AN} = - \overrightarrow {AB} + \dfrac{1}{2}\overrightarrow {AC} = \dfrac{5}{4}\left( { - \dfrac{4}{5}\overrightarrow {AB} + \dfrac{2}{5}\overrightarrow {AC} } \right) = \dfrac{5}{4}\overrightarrow {BI} \end{array}\)
Do đó \(\overrightarrow {BN} = \dfrac{5}{4}\overrightarrow {BI} \) nên ba điểm B,I,N thẳng hàng.
