Question 55
Given a triangle $ABC$, with centroid $G$.
a) Proof that for any point on the $M$ plane the following is true:
$$
\overline{MA}^{2}+\overline{MB}^{2}+\overline{MC}^{2} \geq \overline{GA}^{2}+\overline{GB}^{2}+\overline{GC}^{2}
$$
obtaining equality if and only if $M=G$.
b) Given a number $k\gt \overline{GA}^{2}+\overline{GB}^{2}+\overline{GC}^{2}$, find the locus of the points $M$ such that
$$
\overline{MA}^{2}+\overline{MB}^{2}+\overline{MC}^{2}=k
$$
a) Proof that for any point on the $M$ plane the following is true:
$$
\overline{MA}^{2}+\overline{MB}^{2}+\overline{MC}^{2} \geq \overline{GA}^{2}+\overline{GB}^{2}+\overline{GC}^{2}
$$
obtaining equality if and only if $M=G$.
b) Given a number $k\gt \overline{GA}^{2}+\overline{GB}^{2}+\overline{GC}^{2}$, find the locus of the points $M$ such that
$$
\overline{MA}^{2}+\overline{MB}^{2}+\overline{MC}^{2}=k
$$