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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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Where is this problem from? –  Qiaochu Yuan Oct 14 '12 at 0:51
    
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? –  Haskell Curry Oct 14 '12 at 1:02
    
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. –  Qiaochu Yuan Oct 14 '12 at 1:20
    
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. –  Haskell Curry Oct 14 '12 at 1:50
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up vote 6 down vote accepted

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

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Thanks! You've provided a very nice pancake indeed. –  Haskell Curry Oct 14 '12 at 2:07
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