\(\overrightarrow {AI} = \overrightarrow {AB} + \overrightarrow {BI} = \overrightarrow {AB} + \frac{1}{2}\overrightarrow {BG} = \overrightarrow {AB} + \frac{1}{2}\overrightarrow {AH} \)

\[ = \overrightarrow {AB} + \frac{1}{2}\left( {\overrightarrow {AD} + \overrightarrow {AE} } \right) = \overrightarrow a + \frac{1}{2}\overrightarrow b + \frac{1}{2}\overrightarrow c \].