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Arnol’d, the Jacobi Identity, and Orthocenters

Nikolai V. Ivanov
The American Mathematical Monthly
Vol. 118, No. 1 (January 2011), pp. 41-65
DOI: 10.4169/amer.math.monthly.118.01.041
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Page Count: 25
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Arnol’d, the Jacobi Identity, and Orthocenters
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Abstract The three altitudes of a plane triangle pass through a single point, called the orthocenter of the triangle. This property holds literally in Euclidean geometry, and, properly interpreted, also in hyperbolic and spherical geometries. Recently, V. I. Arnol’d offered a fresh look at this circle of ideas and connected it with the well-known Jacobi identity. The main goal of this article is to present an elementary version of Arnol’d’s approach. In addition, several related ideas, including ones of M. Chasles, W. Fenchel and T. Jørgensen, and A. A. Kirillov, are discussed.

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