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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

  

Source: Abo, Hirotachi - Department of Mathematics, University of Idaho

 

Collections: Mathematics