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Pascal's Theorem


Date: 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/   
    
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Triangles and Other Polygons

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