Problem
GEO-B3-M02-P022 Pappus as a Projective Criterion
B. New Original Problem. On lines \(l\) and \(m\), triples of points \(A,B,C\) and \(A_1,B_1,C_1\) are chosen. Let \(P=AB_1\cap A_1B\) and \(Q=BC_1\cap B_1C\). The line \(PQ\) meets \(AC_1\) at \(R\). Prove that \(R\) lies on the line \(A_1C\).
C. Hint 1. State Pappus so that the third point is \(AC_1\cap A_1C\).
D. Hint 2. Show that the constructed point \(R\) coincides with the third Pappus point.
E. Full Solution.
By Pappus' theorem for points \(A,B,C\) on \(l\) and \(A_1,B_1,C_1\) on \(m\), the points
\[ P=AB_1\cap A_1B,\quad Q=BC_1\cap B_1C,\quad R_0=AC_1\cap A_1C \]
are collinear. The first two are the given points \(P\) and \(Q\), so \(R_0\in PQ\).
But by definition \(R\) is the point \(PQ\cap AC_1\). The point \(R_0\) also lies on \(PQ\) and on \(AC_1\). Hence \(R=R_0\). Therefore \(R\in A_1C\), as required.
This is Pappus used in reverse: instead of proving collinearity, one recovers a missing line.