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."

Consider all the circles that intersect the points (0,1) and (0,-1).  Let C(x) be the circle that crosses these two points along with point (x,0), where x is in the set (0,∞).  We also define C(0) to be the y-axis.  All these circles and the line, taken together, cover the entire plane.

In the bijection, we would need to map (0,1) and (0,-1) to a pair of points that I will call, without loss of generality, (0,1) and (0,-1) respectively.  Also, every circle C(x) must be mapped to a circle C(x') on this new plane.  So we need to bijectively map x to x'; [0,∞) to [0,∞).

If we were to look for a counterexample, there must be at least one line that maps to a non-line.  Without loss of generality, we choose the y-axis, C(0), to be this line.  So in our mapping of x to x', 0 cannot be mapped to 0.  Therefore, there must exist a number c in the set (0,∞) that maps to 0.  But this contradicts our assumption that every circle maps to a circle.  Therefore, no counterexample exists.

I may have made a mistake somewhere on the way.

Edited on November 16, 2006, 8:45 pm

Edited on November 16, 2006, 8:45 pm

Posted by Tristan on 2006-11-16 20:44:19