Problem
GEO-B3-M02-P004 Exceptional Line and Parallelism
B. New Original Problem. A projective transformation of the plane sends a line \(s\) to the line at infinity. Let two ordinary lines \(a\) and \(b\) meet at a point \(T\in s\). Prove that their images are parallel. Also prove the converse: if the images of two lines are parallel, then the intersection point of the original lines lies on \(s\).
C. Hint 1. Recall that the intersection of lines maps to the intersection of their images.
D. Hint 2. Parallel lines meet at a point at infinity.
E. Full Solution.
Since \(T=a\cap b\), its image \(T'\) is the intersection of the images \(a'\) and \(b'\). But \(T\in s\), and \(s\) maps to the line at infinity, so \(T'\) is a point at infinity. Two ordinary lines meeting at a point at infinity are parallel. Hence \(a'\parallel b'\).
Conversely, if \(a'\parallel b'\), then their intersection is a point at infinity. Its preimage lies on the preimage of the line at infinity, namely on \(s\). Therefore \(a\cap b\in s\).
This task supports later normalisations: “send a line to infinity” means replacing intersections on it by parallelism.