We all learn in the introductory mechanics class that the motion of a rigid body can be composed of a rotation and a translation. But how can one prove this? I mean: Let us have some rigid body in two configurations in space, how can I show that I can transform one configuration to another by just a translation and rotation? I could take the two configurations to have the same center of mass or any other point by a translation. But how do I then find the rotation (constructively)?