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 re(2): Full Proof by Tristan)

We have a bijection f that maps the plane P
onto the plane P' that also maps circles into
circles. Per your proof (WLOG) we have 
f(a,b) = (c,d) with f(0,1) = (0,1) and
f(0,-1) = (0,-1). You want to show that f
maps the y-axis to the y-axis. To do this
you must show a=0 <==> c=0. You pick points
(0,0) and (0,2) in the P' plane and derive a
contradiction. I would have derived the
contradiction differently, but I agree that
c=0 ==> a=0. My problem is we still need to
show that a=0 ==> c=0.

Posted by Bractals on 2006-12-17 15:35:08