Pascal's TheoremDate: 10/07/98 at 17:21:04 From: Omar Saleh Subject: Sequential Math 1A Dear Dr. Math, I am doing a report on Pascal's Triangle and Theorem. I got a lot of information on Pascal's Triangle from books and the Internet, but my problem is that I can't find any information about Pascal's Theorem. Could you please give me listings of some Web sites that can help me with my problem or explain Pascal's Theorem to me? I would really appreciate your help. Sincerely, Omar Saleh Date: 10/08/98 at 21:13:36 From: Doctor Santu Subject: Re: Sequential Math 1A Dear Omar: Look at the following image: You should see a circle, six points around the edge of it, and a pattern of lines joining some of the points to other points. Pascal's theorem is about the fact that three of these points where the diagonals cross are collinear, that is, they lie on a line. (Any three random points do not normally lie on the same line. It so happens that if you pick any six points on any circle, and join them the way I have shown, they cross in three lines that do happen to lie on a line.) The actual theorem is more general. A circle is just a special case of a large family of curves including ellipses, hyperbolas, parabolas, and ellipses. For any of these curves, if you pick any six points and join them with 3 pairs of lines, the points of crossing should line up in the so-called Pascal Line. The reverse is also true: Brianchon's Theorem says that if you have a curve such that any six points can be joined to intersect in three collinear points, then the curve has to be a circle, an ellipse, a hyperbola, a parabola, or any curve in the family known as Conic Sections. (Conic Sections are obtained by slicing a double-cone with various planes. A double cone is two perfect cones joined tip-to-tip, like what you would get if you rotated a letter X around a vertical axis. This family of curves also includes pairs of intersecting straight lines.) If you would still like to make an Internet search, look for Brianchon's Theorem, and you might get some references that way. Even better would be to ask a high-school math teacher for a book on geometry. - Doctor Santu, The Math Forum http://mathforum.org/dr.math/ |
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