Homotheties: Let $A$ and $B$ be distinct points of a circle $o$. What is the set of possible centroids of triangles $ABC$ with $C\in o$? Question: Let $A$ and $B$ be distinct points of a circle $o$. What is the set of possible centroids of triangles $ABC$ with $C\in o$?
Here is what I have:
The angle at $C$ will always be the same as it is always subtended by the same arc as $A$ and $B$ are fixed.
There are 2 cases: Either $C$ lies on the small arc of $AB$ or $C$ lies on the big arc of $AB$.
In the case where $C$ lies on the large arc, by looking at the possible positions of $C$ one can observe that at some point $C$ and $A$ are on the same diameter and at another point $C$ and $B$ are on the same diameter. Also, it is worth mentioning that the midpoint at which $c$ intersects on the chord $AB$ does not depend on $C$ and is thus always the same.
One can observe, by determining various points that satisfy the criteria in a drawing, that the possible points $C$ all seem to lie on a smaller circle contained in the original circle $o$. 
One can then guess that the center of this smaller circle is a possible center of homothethy mapping the smaller circle to the larger circle with scale $r_1/r_2$ where $r_1$ is the radius of the larger circle and $r_2$ is the radius of the smaller circle(I am really not sure about this).
I am not sure what to say about the case where $c$ lies on the small arc and am not sure where to continue with the problem.
Any help is appreciated.

Here's a moderately obnoxious idea:
If you use complex numbers and set your circle to be the unit circle, then the centroid of the triangle determined by $a,b,c$ is simply $\frac{a+b+c}{3}$. Thus, the locus of possible centroids, as $A$ and $B$ are fixed and $C$ varies, is simply the circle with radius $\frac{1}{3}$ centered at $\frac{a+b}{3}$ (I think you need to get rid of two of the points because $C\neq A,B$, but oh well). 
There should be a geometric interpretation of this idea as well, if you essentially try to phrase everything with homotheties.
