Problem
GEO-B3-M02-P002 Three Fixed Points
B. New Original Problem. A projective transformation of a line \(l\) fixes three distinct points \(A,B,C\). Prove that it is the identity.
C. Hint 1. Take an arbitrary point \(X\) and compare it with its image.
D. Hint 2. Use the equality \((A B C X)=(A B C X')\).
E. Full Solution.
Let \(X\) be an arbitrary point of the line, and let \(X'\) be its image. Since the transformation is projective, it preserves cross-ratio:
\[ (A B C X)=(A B C X'). \]
For fixed distinct \(A,B,C\), the value \((A B C T)\) uniquely determines the point \(T\) on the line. Hence \(X'=X\). Since \(X\) was arbitrary, every point of the line is fixed.
This fact is used later as a powerful tool: a complicated composition of projections is often proved to be the identity through three fixed points.