Gọi I là trọng tâm tam giác:
\(\Rightarrow\overrightarrow{IA}+\overrightarrow{IB}+\overrightarrow{IC}=\overrightarrow{0}\)
Kẻ đường cao AH
\(\Rightarrow AI=\dfrac{2}{3}AH=\dfrac{2}{3}.\dfrac{a\sqrt{3}}{2}=\dfrac{a\sqrt{3}}{3}\)
\(\Rightarrow AI^2=\dfrac{a^2}{3}=BI^2=CI^2\)
\(MA^2+MB^2+MC^2=\left(\overrightarrow{MI}+\overrightarrow{IA}\right)^2+\left(\overrightarrow{MI}+\overrightarrow{IB}\right)^2+\left(\overrightarrow{MI}+\overrightarrow{IC}\right)^2\) \(\Leftrightarrow2a^2=3MI^2+2\overrightarrow{MI}\left(\overrightarrow{IA}+\overrightarrow{IB}+\overrightarrow{IC}\right)+IA^2+IB^2+IC^2\)
\(\Leftrightarrow2a^2=3MI^2+3IA^2\)
\(\Leftrightarrow2a^2=3MI^2+\dfrac{3.a^2}{3}\)
\(\Leftrightarrow MI^2=\dfrac{a^2}{3}\)
\(\Leftrightarrow MI=\dfrac{a\sqrt{3}}{3}\)
\(\Rightarrow M\in\) đường tròn tâm I bán kính \(\dfrac{a\sqrt{3}}{3}\)
