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

Abu-Saymeh

Hajja

Hayajneh

2012

J. Geom.

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

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

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