Let $ f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} $ be a bijective function. If the image of any circle under $ f $ is a circle, prove that the image of any straight line under $ f $ is a straight line.

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    $\begingroup$ Where is this problem from? $\endgroup$ Oct 14, 2012 at 0:51
  • $\begingroup$ This problem was a question some of my classmates and I discussed over tea-time. We have regular discussions like this. Have you seen it somewhere before? $\endgroup$ Oct 14, 2012 at 1:02
  • $\begingroup$ Mostly I'm wondering if you happen to know for a fact that this is true (e.g. because it was stated in a book of problems somewhere) or just believe it to be true. $\endgroup$ Oct 14, 2012 at 1:20
  • $\begingroup$ I'm not sure if this was obtained from a book. However, someone in my group mentioned that this was a folklore result and that he had seen a proof of it in some article. None of the rest of us could find a proof ourselves, and that fellow had trouble remembering the article where he had seen the proof. $\endgroup$ Oct 14, 2012 at 1:50

1 Answer 1


This has the result (second page). I hope it's thorough enough to placate your curiosity...

  • $\begingroup$ Thanks! You've provided a very nice pancake indeed. $\endgroup$ Oct 14, 2012 at 2:07

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