This exercise is particularly puzzling for me and I cannot seem to get anywhere with it. I am expected to solve this problem by implementing homotheties to circles. The problem states:
Let A and B be distinct points of a circle o. What is the set of possible centroids of triangles ABC with C belonging to o? Recall that the centroid of a triangle is the intersection point of the medians.
All advice is much appreciated!
Hint: let $D$ be the midpoint of $AB$, and $G$ the centroid of $\triangle ABC$. Then $G \in CD$ and $\frac{DG}{DC}=\frac{1}{3}$, so the locus of $G$ is homothetic to the locus of$C$.