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."
I am curious that nobody is addressing the form of f.
Probably I am missing something, but the only bijection functions I can
think of which map all circles into circles result from sucessively
applying some combination of functions which do either (a) translation,
(b) rotation around an arbitrary point, or (c) magnification from an
arbitrary point.
And all of these operations map lines into lines also.
Of course, a good continuity argument might be simpler than a proof of what I am suggesting ...
Not a proof, but ...
