Let $n$ be a positive integer number and $P$ be a point in a plane. Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ points in the plane, we take modulo $m$ for $A_j$ (it is mean $A_{m+i}=A_{i}$ for $i=1, 2, \cdots$).

Now we rotate $P$ clockwise above $A_1$ with angle $\frac{\pi}{n}$ we have point $P_1$, rotate $P_1$ clockwise above $A_2$ with angle $\frac{\pi}{n}$ we have point $P_2$ $,\cdots,$ rotate $P_k$ clockwise above $A_{k+1}$ with angle $\frac{\pi}{n}$ we have point $P_{k+1}$....

By my computation with $(m,n)=(3,1), (3,2), (3,3), (4,3), (3,4), (2,3), (2,4), (2,5)$....I see that exist $k$ such that $P_{k+1}\equiv P$

My question: What condition of $(m,n)$ such that exist $k$ so that $P_{k+1}\equiv P$? Version of this problem for Euclidean space (or non-Euclidean geometries)?