\[\begin{array}{l}
\overrightarrow {AI} = \dfrac{1}{2}\left( {\overrightarrow {AB} + \overrightarrow {AM} } \right) = \dfrac{1}{2}\overrightarrow {AB} + \dfrac{1}{2}\overrightarrow {AM} \\
= \dfrac{1}{2}\overrightarrow {AB} + \dfrac{1}{2}.\dfrac{1}{2}.\overrightarrow {AC} = \dfrac{1}{2}\overrightarrow {AB} + \dfrac{1}{4}\overrightarrow {AC} \\
\overrightarrow {AF} = \overrightarrow {AB} + \overrightarrow {BF} = \overrightarrow {AB} + \dfrac{1}{3}\overrightarrow {BC} \\
= \overrightarrow {AB} + \dfrac{1}{3}\left( {\overrightarrow {AC} - \overrightarrow {AB} } \right) = \dfrac{2}{3}\overrightarrow {AB} + \dfrac{1}{3}\overrightarrow {AC} \\
= \dfrac{4}{3}\left( {\dfrac{1}{2}\overrightarrow {AB} + \dfrac{1}{4}\overrightarrow {AC} } \right) = \dfrac{4}{3}\overrightarrow {AI}
\end{array}\]
Vậy ba điểm A, I, F thẳng hàng.