It's fairly well known that isometries in Euclidean space are composed of only translations, reflections and rotations. However, I'm finding it difficult to locate a proof of that. As usual, it's "intuitively obvious" but formally I'm not sure where to start. Does anyone know of a good reference on geometry that might have one? (My question is actually set in the context of reading about the Galilean group, with every element of that group being a composition of a rotation, translation and motion with uniform velocity. (Arnold's book on classical mechanics.)) Edit: There seems to be a uniqueness aspect to this too. It seems to me to be connected to the direct/semidirect product nature of the Euclidean group.