proof of Pascal’s mystic hexagram
and the vertices of the given hexagram are
(see Remarks below). The equations of the six sides, arranged in opposite pairs, are then
and the three points of intersection of pairs of opposite sides are
is zero. We have
Using (1) we get
A synthetic proof (without coordinates) of Pascal’s theorem is possible with the aid of cross ratios or the related notion of harmonic sets (of four collinear points).
Pascal’s proof is lost; presumably he had only the real affine plane in mind. A proof restricted to that case, based on Menelaus’s theorem, can be seen at http://www.cut-the-knot.org/Curriculum/Geometry/Pascal.shtml#wordscut-the-knot.org.
|Title||proof of Pascal’s mystic hexagram|
|Date of creation||2013-03-22 13:53:02|
|Last modified on||2013-03-22 13:53:02|
|Last modified by||mathcam (2727)|