Originally Posted by
Pinkk A rigid motion of the Euclidean plane is a transformation $\displaystyle f(P)$ of the plane into itself such that $\displaystyle d(f(P),f(Q))=d(P,Q)$.
Would I just write that if you are given a circle $\displaystyle O$ with center $\displaystyle A$ and any arbitrary point on the circle $\displaystyle B$ that is always distance $\displaystyle \alpha$ from the center, then according to the definition of rigid motion, $\displaystyle d(f(A),f(B))=d(A,B)=\alpha$, so any arbitrary point $\displaystyle B'$ on $\displaystyle O'$ is distance $\displaystyle \alpha$ from $\displaystyle A'$, and so $\displaystyle O'$ is a circle?