Example of a Pascal line.
By capturing six points on a cone section like a hexagon, we can find a line, the Pascal line, as the line through the three intersections of pairs of opposite sides of this hexagon with the statement of Pascal.
If the six points are re-assembled into a new hexagon, the three intersections of pairs in the new hexagon overlay are still aligned. If we call the points A, B, C, D, E and F, then there are ( 5 2 ) = 120 {\displaystyle {5 \choose 2}=120} possible arrays that begin with point A. Each possible hexagon is ranked twice as an arrangement, namely clockwise and left-hand read. Thus we reach 60 possible hexagons and up to 60 Pascallines.
There are twenty combinations of three Pascal lines that all three go through one line, such a point is a point of Steiner, as well as sixty three-legged Pascallines, passing all three by one point, by a point of Kirkman.
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