TITLE: A bijection from the plane to itself that takes a circle to a circle must take a straight line to a straight line.
QUESTION [14 upvotes]: 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.

REPLY [6 votes]: This has the result (second page).  I hope it's thorough enough to placate your curiosity...