Problem
GEO-B3-M02-P007 Desargues: Direct Form
B. New Original Problem. Triangles \(ABC\) and \(A_1B_1C_1\) are such that the lines \(AA_1\), \(BB_1\), \(CC_1\) meet at one point \(O\). Let \(P=AB\cap A_1B_1\), \(Q=BC\cap B_1C_1\), and \(R=CA\cap C_1A_1\). Prove that \(P,Q,R\) are collinear.
C. Hint 1. Try changing the perspective so that \(O\) goes to infinity.
D. Hint 2. In the affine picture you get the standard form of Desargues' theorem.
E. Full Solution.
Choose a projective transformation that sends \(O\) to a point at infinity. Then the images of \(AA_1\), \(BB_1\), \(CC_1\) become parallel.
In this affine model, corresponding vertices of the two triangles lie on parallel lines of one direction. In that model the statement is easily checked by coordinates or by Menelaus: the intersections of corresponding sides are collinear.
Since a projective transformation preserves collinearity, the original points \(P,Q,R\) are also collinear.
The theorem may be treated as a module fact, but here the important point is the projective normalisation: the center of perspective is sent to infinity.