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Associated and Skew-Orthologic Simplexes

Leon Gerber
Transactions of the American Mathematical Society
Vol. 231, No. 1 (Jul., 1977), pp. 47-63
DOI: 10.2307/1997867
Stable URL: http://www.jstor.org/stable/1997867
Page Count: 17
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Associated and Skew-Orthologic Simplexes
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Abstract

A set of $n + 1$ lines in $n$-space is said to be associated if every $(n - 2)$-flat which meets $n$ of the lines also meets the remaining line. Two simplexes are associated if the joins of their corresponding vertices are associated. Two simplexes are (skew-)orthologic if the perpendiculars from the vertices of one on the faces of the other are concurrent (associated); it follows that the reciprocal relation holds. In an earlier paper, Associated and Perspective Simplexes, we gave an affine necessary and sufficient condition for two simplexes to be associated that was so easy to apply that extensions to $n$-dimensions of nearly all known theorems, and a few new ones, were proved in a few lines of calculations. In this sequel we take a closer look at some of the results of the earlier paper and prove some new results. Then we give simple Euclidean necessary and sufficient conditions for two simplexes to be orthologic or skew-orthologic which yield as corollaries known results on altitudes, the Monge point and orthocentric simplexes. We conclude by discussing some of the qualitative differences between the geometries of three and higher dimensions.

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