Perhaps there is some simple explanation here. I couldn't find any, especially since the old problem of inscribing an equilateral triangle in a circle flooded every search engine. Therefore, if this is a duplicate in any way, please forgive me.
Suppose you have an equilateral triangle with vertices ${\rm red}$, ${\rm green}$, and ${\rm blue}$. By brute force, it is easy to see that, no matter where we start - say ${\rm red}$ WLOG - if we select the vertices one at a time until there is none left, then we end up going in an oriented path that is as if we traced a circle, starting at ${\rm red}$, that intersects each vertex; we have either
$${\rm red}\to{\rm green}\to{\rm blue}$$
or
$${\rm red}\to{\rm blue}\to{\rm green}.$$
Why?
It seems strange to me, given that the dihedral group $D_3$ of symmetries of an equilateral triangle is the smallest nonabelian group. True: not all of that group is used, so the symmetries at hand correspond to the group $\Bbb Z_3$, which is abelian; but then again the same problem for an $n$-gon for $n>3$ - that of selecting vertices one by one until all are taken - does not always lead to a "circular motion" (so to speak),${}^\dagger$ whereas $\Bbb Z_n$ is abelian.
Please help :)
$\dagger$: Just start with a vertex, say $v$, then go to an adjacent vertex, then go to the other vertex adjacent to $v$, so that, no matter what happens next, the motion of selecting vertices is not circular.
I'm not sure what you mean by your statement that "not all of that group is used". In fact, the full group of symmetries of the 3 element set $\{{\rm R},{\rm B},{\rm G}\}$ is used, this is the symmetric group $S_3$, which is coincidentally isomorphic to the dihedral group $D_3$, so all of the group $D_3$ is also used. The elements of $S_3\approx D_3$ may be listed as
$${\rm RGB, GBR, BRG, RBG, BGR, GRB}.$$
The same problem for an $n$-gon with $n>3$ has a different outcome, because the symmetric group $S_n$ and the dihedral group $D_n$ have different orders: $n! > 2n$ when $n > 3$; arbitrary permutations correspond to $S_n$; and permutations that come from going around the vertices on the $n$-gon in one direction or the other, one at a time, correspond to $D_n$.