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(3): Not a proof, but ... by Tristan)
Tristan:
I agree that you did not fully define a function, and that my comment
is based on intuition, not proof. My intuition is that any
function which maps every circle intersecting (0,1) and (0,-1) into
another circle will map any circle not intersecting (0,1) and (0,-1)
into a non-circle.
Note that every point in the plane except for X = 0, y<> 1 or -1,
is on a circle intersecting (0,1) and (0,-1). So once you do
define how the one circle maps to another, you have defined a function
on virtually every point in the plane.
re(4): Not a proof, but ...
