Definition: A map is said to be conformal if it preserves angles.

1. Let f be a conformal map. Prove that if A, B, and C are not collinear, then f(A), f(B), and f(C) are not collinear. Is this statement true in Neutral Geometry?
2. Can there exist a conformal map from the Euclidean plane E to the hyperbolic plane H? How about from H to E? Prove your answer.
3. True or False: Every similarity from E to itself is conformal.
4. True or False: Every conformal map from a plane to itself is a similarity. Answer this for E, H, and Neutral geometry. Do your answers contradict the statements in http://www.wikipedia.org/wiki/Stereographic_projection?
5. True or False: Every isometry from a plane to itself is bijective. Answer this for E, H, and Neutral geometry.
6. True or False: Every conformal map from a plane to itself is bijective. Answer this for E, H, and Neutral geometry.