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bijective map where images of lines are lines, and preserve parallels

Source: Spanish Mathematical Olympiad 1985 P1

August 2, 2018
geometric transformationgeometrytransformation

Problem Statement

Let f:P→Pf : P\to P be a bijective map from a plane PP to itself such that: (i) f(r)f (r) is a line for every line rr, (ii) f(r)f (r) is parallel to rr for every line rr. What possible transformations can ff be?