Let f be a one-to-one correspondence of the points in a plane. Prove or disprove the following statement:

"If f maps circles to circles, then it maps straight lines to straight lines."

(In reply to Proof by Tristan)

So you reduced the circle-to-circle mapping to a point-to-point mapping by fixing two points, which are also fix points of f. Interesting idea! No problem with your two "without loss of generality" statements, these can be justified by composing our f with suitable orthogonal maps. But although I think you are really close, I am seeing a (minor?) problem with the [0,∞) to [0,∞) bijection:

How can you be sure that the y-axis maps to a circle or a line? I.e. maybe there is no x' such that f(C(0))=C(x')?

Likewise, how can you be sure that for every x'>0 there is a C(x) with x>0 such that f(C(x))=C(x')?

Posted by JLo on 2006-11-17 12:52:15