a) Ta có:

\(2\overrightarrow {IA} + \overrightarrow {IB} + \overrightarrow {IC} = \overrightarrow {MA} + \left( {\overrightarrow {IM} + \overrightarrow {MB} } \right) + \left( {\overrightarrow {IM} + \overrightarrow {MC} } \right)\)

\( = \overrightarrow {MA} + \left( {\overrightarrow {IM} + \overrightarrow {IM} } \right) + \left( {\overrightarrow {MB} + \overrightarrow {MC} } \right)\)

\[ = \overrightarrow {MA} + 2\overrightarrow {IM} = \overrightarrow {MA} + \overrightarrow {AM} \]

\[ = \overrightarrow {MM} = \overrightarrow 0 \]

b) \(2\overrightarrow {IA} + \overrightarrow {IB} + \overrightarrow {IC} = \overrightarrow 0 \)

\( \Leftrightarrow 2\left( {\overrightarrow {IO} + \overrightarrow {OA} } \right) + \left( {\overrightarrow {IO} + \overrightarrow {OB} } \right) + \left( {\overrightarrow {IO} + \overrightarrow {OC} } \right) = \overrightarrow 0 \)

\( \Leftrightarrow \left( {2\overrightarrow {OA} + \overrightarrow {OB} + \overrightarrow {OC} } \right) + \left( {2\overrightarrow {IO} + \overrightarrow {IO} + \overrightarrow {IO} } \right) = \overrightarrow 0 \)

\( \Leftrightarrow \left( {2\overrightarrow {OA} + \overrightarrow {OB} + \overrightarrow {OC} } \right) + 4\overrightarrow {IO} = \overrightarrow 0 \)

\( \Leftrightarrow 2\overrightarrow {OA} + \overrightarrow {OB} + \overrightarrow {OC} = 4\overrightarrow {OI} \)