Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abu-Saymeh, S., Hajja, M.: Equicevian points on the altitudes of a triangle. Elem. Math. (2012, in press)
Berger M.: Geometry II. Springer, New York (1996)
Berele A., Goldman J.: Geometry: Theorems and Constructions. Prentice Hall, New Jersey (2001)
Couderc P., Balliccioni A.: Premier Livre du Tétraèdre. Gauthier-Villars, Paris (1935)
Cresswell, D.: A Treatise on Spherics, Comprising The Elements of Spherical Geometry and of Plane and Spherical Trigonometry. Cambridge (1816)
Crilly T., Millward S.: An optimisation problem for triangles. Math. Gaz. 76, 345–350 (1992)
Dickson L.E.: New First Course in the Theory of Equations. Wiley, New York (1967)
Edmonds A.L., Hajja M., Martini H.: Orthocentric simplices and their centers. Results Math. 47, 266–295 (2005)
Edmonds A.L., Hajja M., Martini H.: Orthocentric simplices and biregularity. Results Math. 52, 41–50 (2008)
Ericksson F.: The law of sines for tetrahedra and n-simplices. Geom. Dedicata 7, 71–80 (1978)
Ericksson F.: On the measure of solid angles. Math. Mag. 63, 184–187 (1990)
Forder H.G.: A School Geometry. Cambridge at the University Press, Cambridge (1949)
Gaddum J.W.: The sums of the dihedral and trihedral angles in a tetrahedron. Amer. Math. Mon. 59, 370–371 (1952)
Hajja M.: Another curious cubic. Math. Gaz. 79, 99–102 (1995)
Hajja M.: The pons asinorum for tetrahedra. J. Geom. 93, 71–82 (2009)
Hajja M.: The pons asinorum in higher dimensions. Studia Sci. Math. Hung. 46, 263–273 (2009)
Hajja, M.: Lack of relative monotonicity among the various measures of trihedral angles. J. Geom. (2012, accepted)
Hajja, M.: The fencing problem: a blend of calculus, geometry, trigonometry, and number theory (2012, preprint)
Hajja, M.: A geotrigonometric solution of a Diophantine equation of Newton (2012, preprint)
Hajja M., Hayajneh M.: The open mouth theorem in higher dimensions. Linear Algebra Appl. 437, 1057–1069 (2012)
Heath T.L.: Euclid—The Thirteen Books of the Elements, Vol. 1. 2nd edn. Dover, New York (1956)
Heath T.L.: Euclid—The Thirteen Books of the Elements, Vol. 3. 2nd edn. Dover, New York (1956)
Klamkin M.S.: Vector proofs in geometry. Amer. Math. Mon. 77, 1051–1065 (1970)
Lob H.: The orthocentric simplex in space of three and higher dimensions. Math. Gaz. 19, 102–108 (1935)
Millman R.S., Parker G.D.: Geometry—A Metric Approach with Models. 2nd edn. Springer, New York (1991)
Prasolov, V.: Problems in solid geometry. http://students.imsa.edu/~tliu/Math/planegeo.pdf
Problem A2: 65th Annual William Lowell Putnam Mathematical Competition. Math. Mag. 78, 76–77 (2005)
Rotman J.: A First Course in Abstract Algebra. Prentice Hall, New Jersey (1996)
Sofair I.: Problem 1515. Math. Mag. 70, 64 (1997)
Sofair I.: Solution. IBID 71, 69–70 (1998)
Todhunter, L.: Spherical Trigonometry. MacMillan and Company, Cambridge and London (1863)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by a research grant from Yarmouk University.
Rights and permissions
About this article
Cite this article
Abu-Saymeh, S., Hajja, M. & Hayajneh, M. The open mouth theorem, or the scissors lemma, for orthocentric tetrahedra. J. Geom. 103, 1–16 (2012). https://doi.org/10.1007/s00022-012-0116-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00022-012-0116-4