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.
Solution 1:
This has the result (second page). I hope it's thorough enough to placate your curiosity...