Problem
GEO-B3-M02-P008 Desargues: Converse Form
B. New Original Problem. For triangles \(ABC\) and \(A_1B_1C_1\), the points \(P=AB\cap A_1B_1\), \(Q=BC\cap B_1C_1\), and \(R=CA\cap C_1A_1\) are collinear. Prove that the lines \(AA_1\), \(BB_1\), \(CC_1\) are concurrent.
C. Hint 1. Think of the duality of the direct Desargues theorem.
D. Hint 2. One may apply direct Desargues in the dual plane or use a projective transformation.
E. Full Solution.
The converse of Desargues is dual to the direct form: if we interchange points and lines, concurrence becomes collinearity and collinearity becomes concurrence.
Apply the dual form of the direct theorem to the two triples of sides \(AB,BC,CA\) and \(A_1B_1,B_1C_1,C_1A_1\). Their corresponding intersections \(P,Q,R\) lie on one line, so the joins of the corresponding vertices in the dual configuration are concurrent.
In the original language this means that \(AA_1\), \(BB_1\), \(CC_1\) pass through one point.
If students are not yet comfortable with duality, this can be solved by coordinates after sending the line \(PQR\) to infinity.