# 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.

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 $$r/3$$.
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).
• In terms of homotheties, the transformation $T:c\mapsto \frac{a+b+c}{3}$ is a homothety with center $\frac{a+b}{2}$ and scale factor $1/3$. So, it maps a circle of radius $r$ and center $d$ to a circle of radius $r/3$ and center $T(d)$. – Eric Wofsey Dec 11 '18 at 7:20