Problem
GEO-B3-M02-P006 Composition of Two Projections
B. New Original Problem. Lines \(l,m,n\) are pairwise distinct. First a point \(X\in l\) is projected from a center \(O\) to the line \(m\), and then the obtained point is projected from a center \(P\) to the line \(n\). Prove that the resulting map \(l\to n\) is projective.
C. Hint 1. Break the map into two known types.
D. Hint 2. A composition of projective maps is projective.
E. Full Solution.
The central projection from \(l\) to \(m\) with center \(O\) is a projective map. Similarly, the central projection from \(m\) to \(n\) with center \(P\) is projective.
The resulting map \(l\to n\) is the composition of these two maps. By the definition and basic properties of projective maps, a composition of projective maps is projective.
The task prepares students for more complicated compositions: circle - line - circle - line.