 
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.
