Let $m$ be an isometry in euclidean plane that changes orientation. Prove that $m \circ m$ is a translation.I do not have an idea how to start the proof of this exercise.
The orientation-reversing isometries of the Euclidean plane are the reflections and glide reflections. If you apply a reflection twice, you get back the original shape so the translation is trivial. Now, what happens in the other case?