Let
f be a
onetoone 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 yaxis to the yaxis.
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 noncollinear 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 20061218 12:53:22 