Journal of Geometry

, Volume 103, Issue 1, pp 1–16 | Cite as

The open mouth theorem, or the scissors lemma, for orthocentric tetrahedra

  • Sadi Abu-Saymeh
  • Mowaffaq HajjaEmail author
  • Mostafa Hayajneh


Propositions 24 and 25 of Book I of Euclid’s Elements state the fairly obvious fact that if an angle in a triangle is increased (without changing the lengths of its arms), then the length of the opposite side increases. In less technical terms, the wider you open your mouth, the farther apart your lips are. In this paper, we see that this has a very satisfactory analogue for orthocentric (but not for general) tetrahedra.

Mathematics Subject Classification (2010)

52B10 51M04 


Content of a solid angle open mouth theorem orthocentric tetrahedron polar sine sine of a solid angle solid angle trihedral angle 


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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Sadi Abu-Saymeh
    • 1
  • Mowaffaq Hajja
    • 1
    Email author
  • Mostafa Hayajneh
    • 1
  1. 1.Mathematics DepartmentYarmouk UniversityIrbidJordan

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