# A bijection from the plane to itself that takes a circle to a circle must take a straight line to a straight line.

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