Problem
GEO-B3-M02-P021 General Pascal from a Special Case
B. New Original Problem. Assume Pascal's theorem has already been proved for a circle in the case where one pair of opposite sides of the hexagon is parallel. Explain how to derive Pascal's theorem for arbitrary six points \(A,B,C,D,E,F\) on one conic.
C. Hint 1. Use a projective transformation.
D. Hint 2. Send one of the three intersections of opposite sides to infinity.
E. Full Solution.
Let \(P=AB\cap DE\), \(Q=BC\cap EF\), and \(R=CD\cap FA\). Choose a projective transformation that sends the conic to a circle and sends \(P\) to a point at infinity. Then the images of the lines \(AB\) and \(DE\) become parallel.
In the new configuration, apply the proved special case of Pascal: the images of \(Q\) and \(R\) lie on a line parallel to the images of \(AB\) and \(DE\), so this line passes through the image of \(P\).
Therefore the images of \(P,Q,R\) are collinear. Since projective transformations preserve collinearity, the original points \(P,Q,R\) are also collinear.
This is a methodological task: it shows why sending a point to infinity is useful.