Problem
GEO-B3-M02-P024 Closure of a Projection Chain
B. New Original Problem. Lines \(l_1,l_2,\ldots,l_n\) and points \(O_1,O_2,\ldots,O_n\) are given. Starting from \(X_1\in l_1\), construct a chain by \(X_{i+1}=O_iX_i\cap l_{i+1}\), where \(l_{n+1}=l_1\). It is known that for three distinct starting points \(X_1\), the chain returns to the starting point after \(n\) steps. Prove that this is true for every starting point \(X_1\in l_1\) for which all constructions are defined.
C. Hint 1. Consider the resulting self-map of the line \(l_1\).
D. Hint 2. It is projective and has three fixed points.
E. Full Solution.
Each step \(X_i\mapsto X_{i+1}\) is a central projection from the line \(l_i\) to the line \(l_{i+1}\) with center \(O_i\). Thus each step is a projective map between lines.
The composition of all \(n\) steps is a projective transformation of the line \(l_1\) to itself. By assumption, this transformation has three distinct fixed points: the three starting points for which the chain closes.
A projective transformation of a line with three fixed points is the identity. Therefore the resulting map sends every point \(X_1\), for which all intermediate projections are defined, back to itself. Hence the chain closes for every such starting point.
This is one of the most important olympiad schemes: “if it closes three times, it always closes”, when the process is governed by a projectivity of a line.