How to prove that the pointreflection at the midpoint of two several points $A,B\in\mathfrak{L}$ in a regular pointlattice $\mathfrak{L}$ fix the lattice $\mathfrak{L}$?
We call $\mathfrak{L}\subset\mathbb{E}$ a regular pointlattice in the euclidian plane $\mathbb{E}$ if and only if:
$$\exists A,B,C\in\mathfrak{L}\text{ non collinar}:\mathfrak{L}=\{\tau_{A,B}^i\circ\tau_{A,C}^j(A)\mid i,j\in\mathbb{Z}\}$$
in wich $\tau_{A,B},\tau_{A,C}\in Iso(\mathbb{E})$ are translations from Point $A$ to $B$ and accordingly $A$ to $C$.
my attampt...
proof: Let $A,B$ be two serveral points in the pointlattice $\mathfrak{L}$.
Further more there is an arbitray point $P\in\mathbb{E}$. If the midpoint $MP[A,B]$ lies in $\mathfrak{L}$: $\exists n,m\in\mathbb{Z}:P=\tau_1^m\circ\tau_2^n(MP[A,B])$, while $\tau_1,\tau_2\in Iso(\mathbb{E})$ are generators of generating elemnts of the regular pointlattice. It is easy to see, that $\sigma_{MP[A,B]}(A)=\tau_1^{-m}\circ\tau_2^{-n}(MP[A,B])$.
But what if $MP[A,B]\notin\mathfrak{L}$?
It seems easier just to say that $\mathcal{L} = \{na+mb:n,m\in\mathbb{Z}\}$ for fixed linearly independent $a, b\in\mathbb{E}$. What is $\tau_{A,B}$ if not a function which has $\tau_{A,B}(x)=x+(B-A)$ for all $x\in\mathbb{E}$?
Wolog one of the points is $(0,0)$; let the other be $na+mb$. The midpoint is ${1\over 2}(na+mb)$. The reflection $r$ in that midpoint takes $z\in\mathbb{E}$ to $r(z)={1\over 2}(na+mb)-(z-{1\over 2}(na+mb))=$ $na+mb-z$, so if $z=n'a+m'b\in\mathcal{L}$, then $r(z)=(n-n')a+(m-m')b\in\mathcal{L}$, as we wished to prove.