Problem Set 15 Problem 1. Let H be a hexagon. The intersections of the opposite sides of H Summary: Problem Set 15 Problem 1. Let H be a hexagon. The intersections of the opposite sides of H determine 3 points A, B, C. Prove that if A, B, C lie on a line then the vertices of the hexagon lie on a conic. Problem 2. Let C be an irreducible curve of degree 2 in P2 . Prove that the dual of C is a conic. Problem 3. State the dual form of Pascal's theorem. Problem 4. State the dual form of Pappus's theorem. Problem 5. Give a brief explanation of why there is not a conic passing through 6 general points in P2 . Problem 6. Let P1, P2, . . . , P6 be 6 points on an irreducible curve, C, of degree 2 in P2 . Let Lij denote the line passing through points Pi and Pj. Let Q1 = L12 L34, Q2 = L23L45, Q3 = L34L56, Q4 = L45L61, Q5 = L56L12, Q6 = L16L23. Prove that there is a conic passing through Q1, . . . , Q6. Problem 7. Let C and D be curves in P2 of degree n. Suppose C and D meet in exactly n2 Collections: Mathematics