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(4): Full Proof by Joel)

Hi Joel. Thanks for the input.

I think my confusion stems from the mapping
of the y-axis to the y-axis. 

Let P and P' be abstract planes and A and B
distinct points in P.

We want to show that 

     f[Line(A,B)] = Line(f(A),f(B))

                  or

     Line(A,B) = g[Line(f(A),f(B))

Note: I am using g for f^(-1).

I will stipulate that 

     Line(f(A),f(B)) is a subset of f[Line(A,B)].

Could you please rewrite your proof so that

Points are denoted by A, B, C, ... or 
                      f(A), f(B), f(C), ... or
                      g(A), g(B), g(C), ...

A circle is denoted by Circle(X,Y,Z) where
  X, Y, and Z are non-collinear points. Hopefully,
  from the points I can determine whether the
  circle is in P or P'.

Note: If Circle(X,Y,Z) is in P, then f[Circle(X,Y,Z)]
      = Circle(f(X),f(Y),f(Z)) is a circle in P'.

Note: If Circle(X,Y,Z) is in P', then g[Circle(X,Y,Z)]
      might or might not equal Circle(g(X),g(Y),g(Z))
      in P.

Posted by Bractals on 2006-12-18 12:53:22