By Derrick Norman Lehmer
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Additional info for An Elementary Course in Synthetic Projective Geometry
The result of the preceding paragraph may then be stated as follows: Given three points, A, B, C, in a straight line, if a pair of opposite sides of a complete quadrangle pass through A, and another pair through C, and one of the remaining two sides goes through B, then the other of the remaining two sides will go through a fixed point which does not depend on the quadrangle employed. 29. Four harmonic points. Four points, A, B, C, D, related as in the preceding theorem are called four harmonic points.
The above theorem, which is of cardinal importance in the theory of the point-row of the second order, is due to Pascal and was discovered by him at the age of sixteen. It is, no doubt, the most important contribution to the theory of these loci since the days of Apollonius. If the six points be called the vertices of a hexagon inscribed in the curve, then the sides 12 and 45 may be appropriately called a pair of opposite sides. Pascal's theorem, then, may be stated as follows: The three pairs of opposite sides of a hexagon inscribed in a point-row of the second order meet in three points on a line.
Cut across this set of planes by another plane not passing through S. This plane cuts out on the set of seven planes another quadrangle which determines four new harmonic points, A', B', C', D', on the lines joining S to A, B, C, D. But S may be taken as any point, since the original quadrangle may be taken in any plane through A, B, C, D; and, further, the points A', B', C', D' are the intersection of SA, SB, SC, SD by any line. We have, then, the remarkable theorem: 32. If any point is joined to four harmonic points, and the four lines thus obtained are cut by any fifth, the four points of intersection are again harmonic.
An Elementary Course in Synthetic Projective Geometry by Derrick Norman Lehmer
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