Journal of Geometry

, Volume 103, Issue 1, pp 1-16

First online:

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

  • Sadi Abu-SaymehAffiliated withMathematics Department, Yarmouk University
  • , Mowaffaq HajjaAffiliated withMathematics Department, Yarmouk University Email author 
  • , Mostafa HayajnehAffiliated withMathematics Department, Yarmouk University

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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