Problem
GEO-B3-M02-P019 A Cyclic Projectivity of Order Three
B. New Original Problem. A projective transformation \(f\) of a line \(l\) sends three distinct points \(A,B,C\) as follows: \(f(A)=B\), \(f(B)=C\), \(f(C)=A\). Prove that \(f^3\) is the identity transformation of \(l\).
C. Hint 1. Look at what \(f^3\) does to \(A,B,C\).
D. Hint 2. Use the fact about three fixed points.
E. Full Solution.
Applying \(f\) three times, we get:
\[ f^3(A)=A,\quad f^3(B)=B,\quad f^3(C)=C. \]
The map \(f^3\) is a composition of projective transformations, so it is projective. It has three distinct fixed points \(A,B,C\). Therefore, by the basic fact about projectivities of a line, \(f^3\) is the identity.
This problem looks algebraic, but it prepares for geometric situations where \(f\) arises as a composition of projections.