I was experimenting with GeoGebra and I found something that I don't quite understand.
I wanted to see what would happen to the centroid of a triangle if one of its sides rotated inside a circle ($\bar{AB}$ would always be the diameter).
But when I rotated the side, the centroid stayed in place and didn't move at all.
I tried to generalize this so I could understand what was happening: I started by defining an arbitrary curve $\vec{r}(t) = r_x \vec{i} + r_y \vec{j}$ and a point $A=\vec{r}(t_A)$ on that curve.
Because I want $\bar{AB}$ to be the diameter, Point $B$ would have to be $B=-\vec{r}(t_A)$.
I constructed $\overset{\triangle}{ABC}$ with points $A$, $B$ and $C=(x_c,y_c)$, then I found the medians and the centroid of the triangle by solving for where they intersect.
Interestingly, I found that the centroid is always $G=(\frac{x_c}{3},\frac{y_c}{3})$; and because I chose an arbitrary curve for points $A$ and $B$ to move on, the centroid would always have these coordinates and will only be dependent on point $C$ no matter what the curve is. This also works in 3 dimensions.
I don't know if it's possible to upload videos on here, but seeing this in motion is really interesting so here's the Google Drive link to the GeoGebra files: Link
I understand why $G$ stays in place from a mathematical standpoint because I solved for it; but I don't quite get the intuition behind it and I haven't seen something like this before, why does it have to stay in place no matter what the curve is?
Also, I know this is a place for math questions and I'm sorry if this last part is off-topic, but would this have anything to do with the first animation on the Wikipedia page for the Three-Body Problem? The fact that the center of mass of these 3 particles remains in place seems kind of similar to this question, is this a physical thing or a geometrical thing?