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Concurrence of tetrahedral cevians associated with triangle centers

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Abstract

It is well known that the center of mass of a tetrahedron is the intersection of the line segments that join its vertices to the centers of mass of the opposite faces, and that a similar statement holds for simplices in higher dimensions. This paper addresses the question whether this nice inductive property of the center of mass also holds for other centers. It proves that the center of mass is the only center that has this property, and investigates in detail the situation for the other well known centers, namely, the Gergonne center, the Nagel center, the Lemoine center, the incenter, the orthocenter, the Fermat-Torricelli center, and the circumcenter. Some questions for further research are raised.

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Correspondence to M. Hajja.

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Hajja, M., Hammoudeh, I., Hayajneh, M. et al. Concurrence of tetrahedral cevians associated with triangle centers. J. Geom. 111, 8 (2020). https://doi.org/10.1007/s00022-019-0513-z

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Keywords

  • Centroid
  • Cevian
  • circumcenter
  • circumscriptible simplex
  • Fermat–Torricelli point
  • Gergonne center
  • incenter
  • isodynamic simplex
  • isogonic simplex
  • Lemoine center
  • Nagel center
  • orthocenter
  • orthocentric simplex
  • simplex

Mathematics Subject Classification

  • Primary 52B10
  • Secondary 52B12
  • 52B15
  • 51M20
  • 52B11
  • 51N20