TITLE: Classification of Euclidean plane isometries
QUESTION [6 upvotes]: I suppose this question has already been asked here, but I cannot find it. 
Is there any simple way to prove that there are 5 possibilities for isometries in the Euclidean plane? Namely: Identity, Reflection, Rotation, Translation and  Glide reflection. 
Articles I've read (including wikipedia) simply say that it can be proved, but I am curious about how to prove that. Could you recommend me a book, etc where this subject is described? Or post the proof here?

REPLY [4 votes]: Any isometry can be fully described by its effect on three points, unless these lie on a single line. This is the case because for any given point, knowing the distances to the three defining preimage points will define three circles of corresponding radius about the image points. Two of them will intersect in two points, and the third will select one of these two points of intersection. So now you have some tool to describe all isometries.
Next you get every image point in place one after the other. You can map the first point using a translation, preserving the relative position of the second and third point in the process. Next you can map the second point using a rotation around the image of the first, since its distance to the first point is preserved. Unless the third point is already in its final position, you can use a reflection in the axis spanned by the first two image points to get it there. So taken together, any isometry can be described by a translation (perhaps by a zero displacement), followed by a rotation (perhaps by $0°$) possibly followed by a reflection.
Now all you have to do is analyze the possible ways these three steps can interact. A non-zero translation and a non-zero rotation together are simply a rotation around some other center. A rotation followed by a reflection is a reflection in some different line. A translation followed by a reflection in an axis parallel to the direction of the translation is a glide reflection. If the axis of reflection is not parallel to the direction of the translation, the combination is a reflection in a different axis. And if all steps are identity transformations, then so is the result.