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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
proof of Pascal's mystic hexagram
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(Proof)
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We can choose homogeneous coordinates $(x,y,z)$ such that the equation of the given nonsingular conic is $yz+zx+xy=0$ , or equivalently \begin{equation} \label{eq:conic} z(x+y)=-xy \end{equation}and the vertices of the given hexagram $A_1A_5A_3A_4A_2A_6$ are
| $A_1=(x_1,y_1,z_1)$ |
$A_4=(1,0,0)$ |
| $A_2=(x_2,y_2,z_2)$ |
$A_5=(0,1,0)$ |
| $A_3=(x_3,y_3,z_3)$ |
$A_6=(0,0,1)$ |
(see Remarks below). The equations of the six sides, arranged in opposite pairs, are then
| $A_1A_5: x_1z=z_1x$ |
$A_4A_2: y_2z=z_2y$ |
| $A_5A_3: x_3z=z_3x$ |
$A_2A_6: y_2x=x_2y$ |
| $A_3A_4: z_3y=y_3z$ |
$A_6A_1: y_1x=x_1y$ |
and the three points of intersection of pairs of opposite sides are $$A_1A_5\cdot A_4A_2 = (x_1z_2,z_1y_2,z_1z_2)$$ $$A_5A_3\cdot A_2A_6 = (x_2x_3,y_2x_3,x_2z_3)$$ $$A_3A_4\cdot A_6A_1 = (y_3x_1,y_3y_1,z_3y_1)$$ To say that these are collinear is to say that the determinant
is zero. We have
| $D=$ |
$x_1y_1y_2z_2z_3x_3-x_1y_1z_2x_2y_3z_3$ |
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$+z_1x_1x_2y_2y_3z_3-y_1z_1x_2y_2z_3x_3$ |
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$+y_1z_1z_2x_2x_3y_3-z_1x_1y_2z_2x_3y_3$ |
Using (![[*]](http://images.planetmath.org:8080/cache/objects/4627/js//usr/share/latex2html/icons/crossref.png "[*]") ) we get $$(x_1+y_1)(x_2+y_2)(x_3+y_3)D=x_1y_1x_2y_2x_3y_3S$$ where $(x_1+y_1)(x_2+y_2)(x_3+y_3)\ne 0$ and \begin{eqnarray*} S&= &(x_1+y_1)(y_2x_3-x_2y_3) \\ & +&(x_2+y_2)(y_3x_1-x_3y_1) \\ & +&(x_3+y_3)(y_1x_2-x_1y_2) \\ &= &0 \end{eqnarray*}QED.
Remarks:For more on the use of coordinates in a projective plane, see e.g. Hirst (an 11-page PDF).
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 cut-the-knot.org.
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"proof of Pascal's mystic hexagram" is owned by mathcam. [ full author list (2) | owner history (1) ]
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Cross-references: Menelaus' theorem, affine plane, real, harmonic, Pascal's theorem, proof, projective plane, coordinates, QED, determinant, collinear, opposite sides, intersection, points, sides, conic, nonsingular, equation, homogeneous coordinates
This is version 2 of proof of Pascal's mystic hexagram, born on 2003-08-20, modified 2006-08-17.
Object id is 4627, canonical name is ProofOfPascalsMysticHexagram.
Accessed 3880 times total.
Classification:
| AMS MSC: | 51A05 (Geometry :: Linear incidence geometry :: General theory and projective geometries) |
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Pending Errata and Addenda
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