Let $A,A'$ two affine subspaces of a finite Euclidean Vectorspace $V$. Let $p,p'$ two points, such that $d(A,p)=d(A',p')$. $\dim(A)=\dim(A')$
I would like to show that there exists a movement $\alpha:V->V$ such that $\alpha(A)=A'$ and $\alpha(p)=p'$
I only know, if $\alpha$ is a movement, then the following properties hold:
(2) $\alpha$ is affine
(3) $\alpha$ affine and for a basis $a_1,..,a_n$ it holds that $d(\alpha(a_i),\alpha(a_j))=d(a_i,a_j)$
How can I use this to prove the statement above ?