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

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.

Answer 1

[This is just a translation of Carl Schildkraut's answer into synthetic language.]

Note that the centroid of $ABC$ can be constructed by taking the midpoint $M$ of $AB$ and then taking the point $G$ which is $1/3$ of the way along $MC$ (closer to $M$). This means exactly that it is the image of $C$ under the homothety $T$ with center $M$ and scale factor $1/3$. Since homotheties preserve circles and their centers, this homothety maps a circle with center $D$ and radius $r$ to a circle with center $T(D)$ and radius $r/3$.

Answer 2

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.