$V$ is a finite euclidean vectorspace and $\sigma:V->V$ is a motion, this means that $d(\sigma(a_i),\sigma(a_j))=d(a_i,a_j)$ for an affine coordinate system $a_0,...,a_n$
I know the following two things: $\sigma^2=id_V$ and $A:=\{v\in V:\sigma(v)=v\}$
I already proved that A is an affine subspace, but I would like to know why A is non-empty.
My second question is, why $\sigma$ is identical to a reflection $\sigma_A(v):=\sigma_{V_A}(v-a)+a$ where $\sigma_{V_A}$ is an orthogonal reflection on $V_A$ and $a\in A$ is arbitrary.