MA² + MB² + MC² = \({\overrightarrow {MA} ^2} + {\overrightarrow {MB} ^2} + {\overrightarrow {MC} ^2}\)

= \({\left( {\overrightarrow {MG} + \overrightarrow {GA} } \right)^2} + {\left( {\overrightarrow {MG} + \overrightarrow {GB} } \right)^2} + {\left( {\overrightarrow {MG} + \overrightarrow {GC} } \right)^2}\)

= \[3\overrightarrow {MG} + 2\overrightarrow {MG} .\left( {\overrightarrow {GA} + \overrightarrow {GB} + \overrightarrow {GC} } \right) + {\overrightarrow {GA} ^2} + {\overrightarrow {GB} ^2} + {\overrightarrow {GC} ^2}\]

Ta có tính chất trọng tâm tam giác: \[\overrightarrow {GA} + \overrightarrow {GB} + \overrightarrow {GC} = \overrightarrow 0 \]

Suy ra: \[\overrightarrow {MG} \left( {\overrightarrow {GA} + \overrightarrow {GB} + \overrightarrow {GC} } \right) = \overrightarrow 0 \]

Suy ra: MA² + MB² + MC² = \[3\overrightarrow {MG} + {\overrightarrow {GA} ^2} + {\overrightarrow {GB} ^2} + {\overrightarrow {GC} ^2}\]

Vậy MA² + MB² + MC² = 3MG² + GA² + GB² + GC².