Problem Set 13 Problem 1. It is a well known theorem that the diagonals of a parallelogram Summary: Problem Set 13 Problem 1. It is a well known theorem that the diagonals of a parallelogram bisect each other. If A, B, C, D are the vertices of the parallelogram and if M is the point where the diagonals intersect then the theorem can be shown to be true by showing that the condition (M1 -C1)2 +(M2 -C2)2 -(M1 -B1)2 -(M2 -B2)2 = 0 (and a similar equation for the other diagonal) is forced by the assumptions of the problem. This is essentially an ideal membership question. a) The following conditions of the problem are clear: AB CD, AC BD, A, M, D are collinear, B, M, C are collinear. Are any other conditions needed? b) Now put all the conditions on the computer and prove the theorem compu- tationally. Pascal's Theorem says the following: Theorem 2. Let C be an irreducible conic. Let P1, P2, . . . , P6 be 6 points on the conic. Let A be the intersection of P1P2 and P4P5, B be the intersection of P2P3 and P5P6, and C be the intersection of P3P4 and P6P1. Then A, B, C are collinear. Collections: Mathematics