Problem
GEO-B3-M02-P015 Pappus via a Degenerate Conic
B. New Original Problem. Points \(A,B,C\) lie on a line \(l\), and \(A_1,B_1,C_1\) lie on a line \(m\). Let \(P=AB_1\cap A_1B\), \(Q=AC_1\cap A_1C\), and \(R=BC_1\cap B_1C\). Prove that \(P,Q,R\) are collinear.
C. Hint 1. View the pair of lines \(l\cup m\) as a degenerate conic.
D. Hint 2. Apply Pascal to a suitable order of the six points.
E. Full Solution.
The union \(l\cup m\) of two lines may be regarded as a degenerate conic. Take the hexagon with vertices \(A,B_1,C,A_1,B,C_1\) on this conic.
The pairs of opposite sides give exactly the points \(P,Q,R\), up to the naming order. By Pascal's theorem for a degenerate conic, these three points are collinear.
If the class is not ready for degenerate conics, Pappus can be proved separately by coordinates and then connected to Pascal.