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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.

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    $\begingroup$ A good way to start is to find all orientation reversing isometries. $\endgroup$ – Amitai Yuval Dec 12 '14 at 20:22
  • $\begingroup$ I'm sorry, what is $m*2$? $\endgroup$ – Jorge Fernández Hidalgo Dec 12 '14 at 20:54
  • $\begingroup$ It is m to power 2 $\endgroup$ – mathstudent Dec 12 '14 at 21:07
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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?

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