\(\begin{array}{l}\overrightarrow {AA'}  + \overrightarrow {BB'}  + \overrightarrow {CC'} \\ = \left( {\overrightarrow {AG}  + \overrightarrow {GG'}  + \overrightarrow {G'A'} } \right) + \left( {\overrightarrow {BG}  + \overrightarrow {GG'}  + \overrightarrow {G'B'} } \right) + \left( {\overrightarrow {CG}  + \overrightarrow {GG'}  + \overrightarrow {G'C'} } \right)\\ = \left( {\overrightarrow {AG}  + \overrightarrow {BG}  + \overrightarrow {CG} } \right) + \left( {\overrightarrow {G'A'}  + \overrightarrow {G'B'}  + \overrightarrow {G'C'} } \right) + 3\overrightarrow {GG'} \\ =  - \left( {\overrightarrow {GA}  + \overrightarrow {GB}  + \overrightarrow {GC} } \right) + \left( {\overrightarrow {G'A'}  + \overrightarrow {G'B'}  + \overrightarrow {G'C'} } \right) + 3\overrightarrow {GG'} \\ = \overrightarrow 0  + \overrightarrow 0  + 3\overrightarrow {GG'}  = 3\overrightarrow {GG'} \\ \Rightarrow \overrightarrow {AA'}  + \overrightarrow {BB'}  + \overrightarrow {CC'}  = 3\overrightarrow {GG'} \\ \Rightarrow \left| {\overrightarrow {AA'}  + \overrightarrow {BB'}  + \overrightarrow {CC'} } \right| = 3GG'\end{array}\)

Mà

\(\begin{array}{l}\left| {\overrightarrow {AA'}  + \overrightarrow {BB'}  + \overrightarrow {CC'} } \right| \le \left| {\overrightarrow {AA'}  + \overrightarrow {BB'} } \right| + \left| {\overrightarrow {CC'} } \right|\\ \le \left| {\overrightarrow {AA'} } \right| + \left| {\overrightarrow {BB'} } \right| + \left| {\overrightarrow {CC'} } \right| = AA' + BB' + CC'\end{array}\)

Vậy \(AA' + BB' + CC' \ge 3GG'\)