Associated Topics || Dr. Math Home || Search Dr. Math

### 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

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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search