Two centroids

Let $M$ be the midpoint of $\overline{BC}$ and $O$ be the circuncenter. Let $P$ be the point in $\overline{OM}$ that satisfy $\overline{OP}=2\overline{PM}$ . Then, it is easy to prove that the locus of the centroids of the triangles that share $\overline{BC}$ and the circumcircle is a circunference with radius $\frac{1}{3}R$ and center $P$ . So $M=\frac{2}{3}R$ .

P.S.: let $G$ be the centroid, then it is clear that the triangles $AMO$ and $MPG$ are proporcional in ratio $\frac{1}{3}$