Pascal's Mystic Hexagon

Pascal discovered that if hexagon ABCDEF is inscribed in a conic section, the points of intersection RST of the pairs of opposite chords are collinear. You can generate the conic through points A, B, C, D, E, since F can be derived from an arbitrary line AX through A. Moving X and F traces the conic. Move A, B, C, D, or E to get a new conic that passes through them.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Pascal (1640) discovered that if a hexagon (or hexagram) ABCDEF is inscribed in a conic section, then taking opposite chords, the points of intersection R = AB.ED, S = AF.CD, and T = EF.CB are collinear. (If you put the points in the order AECDBF, as in snapshot 2, then the chords are pairs of opposite sides.) Pascal stated his result slightly differently. In effect, his theorem asserts that AF, CD, and RT run through the same point (S). Braikenridge and Maclaurin (1733) discovered how to generate the conic through A, B, C, D, E. The point F can be derived from an arbitrary line AX through A from the following intersections:
R = AB.ED,
S = AX.CD,
T = RS.CB,
F = AX.ET.
Moving X and F traces the conic. You can choose whether just to trace the conic, have the whole conic displayed, or both.
Pascal's "Essai pour les coniques," written in 1640 when he was sixteen, from D. E. Smith, A Source Book in Mathematics, London: Dover, 1959 pp. 326-330.
H. S. M. Coxeter, Introduction to Geometry, 2nd ed., Wiley, 1989.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+