Abstract
Proposition 5 of Book I of Euclid’s Elements, better known as the Pons Asinorum or the Bridge of Asses, and its converse, Proposition 6, state that two sides of a triangle are equal if and only if the opposite angles are equal. A satisfactory generalization of this statement to general tetrahedra and a strong generalization to orthocentric tetrahedra are obtained in Hajja (J Geom 93:71–82, 2009a), and generalizations to orthocentric d-simplices, d ≥ 3, are obtained in Hajja (Stud Sci Math Hungarica 46:263–273, 2009b). In this paper, the strong generalization is seen to hold for a fairly large family of tetrahedra that are referred to as distinguished. These include acute and rectangular orthocentric tetrahedra as well as circumscriptible and isogonic tetrahedra. It is also seen that the strong generalization fails for isodynamic tetrahedra and that even a weak generalization fails for simplices in dimension four.
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References
Abu-Saymeh S., Hajja M.: The Archimedean arbelos in 3-space. Results Math. 52, 1–16 (2008)
Andreescu T., Enescu B.: Mathematical Olympiad Treasures. Birkhäuser, Boston (2004)
Arnold, V. I. (ed): Arnold’s Problems. Springer, Berlin, New York (2005)
Bentin J.: Regular simplicial distances. Math. Gaz. 79, 106 (1995)
Berger M.: Geometry I. Springer-Verlag, Berlin (1994)
Bradley C.J.: Challenges in Geometry. Oxford University Press, Oxford (2005)
Brown B.H.: A theorem on isogonal tetrahedra. Amer. Math. Mon. 31, 371–375 (1924)
Couderc P., Balliccioni A.: Premier Livre du Tétraèdre. Gauthier-Villars, Paris (1935)
Court N.A.: A special tetrahedron. Amer. Math. Mon. 56, 312–315 (1949)
Court N.A.: A special tetrahedron. Amer. Math. Mon. 57, 176–177 (1950)
Court N.A.: Modern Pure Solid Geometry. Chelsea Publishing Co., New York (1964)
Coxeter H.S.M.: Introduction to Geometry, 2nd edn. Wiley, New York (1969)
Durrell C.V., Robson A.: Advanced Trigonometry. G. Bell and Sons, Ltd., London (1961)
Edmonds A.L., Hajja M., Martini H.: Coincidences of simplex centers and related facial structures. Beit. Algebra Geom. 46, 491–512 (2005a)
Edmonds A.L., Hajja M., Martini H.: Orthocentric simplices and their centers. Results Math. 47, 266–295 (2005b)
Edmonds A.L., Hajja M., Martini H.: Orthocentric simplices and biregularity. Results Math. 52, 41–50 (2008)
Eriksson F.: The law of sines for tetrahedra and n-simplices. Geom. Dedicata 7, 71–80 (1978)
Eriksson F.: On the measure of solid angles. Math. Mag. 63, 184–187 (1990)
Gardner M.: Mathematical Circus. Knopf, New York (1979)
Hajja M.: An advanced calculus approach to finding the Fermat point. Math. Mag. 67, 29–34 (1994)
Hajja M.: Triangle centres: some questions in Euclidean geometry. Internat. J. Math. Ed. Sci. Tech. 32, 21–36 (2001a)
Hajja M.: A vector proof of a theorem of Bang. Amer. Math. Mon. 108, 562–564 (2001b)
Hajja M.: A trigonometric identity that G. S. Carr missed and a trigonometry proof of a theorem of Bang. Math. Gaz. 85, 293–296 (2001c)
Hajja M.: Coincidences of centers in edge-incentric, or balloon, simplices. Results Math. 49, 237–263 (2006)
Hajja M.: The pons asinorum for tetrahedra. J. Geom. 93, 71–82 (2009a)
Hajja M.: The pons asinorum in higher dimensions. Stud. Sci. Math. Hungarica 46, 263–273 (2009b)
Hajja, M.: The failure of the strong pons asinorum theorem for isodynamic tetrahedra, preprint
Hajja, M., Hayajneh, M.: The failure of the weak pons asinorum theorem in dimension four, preprint
Hajja M., Saidi F.: A note on Bang’s theorem on equifacial tetrahedra. J.Geom. Graph. 8, 163–169 (2004)
Hajja M., Walker P.: Equifacial tetrahedra. Internat. J. Math. Ed. Sci. Tech. 32, 501–508 (2001)
Hajja M., Walker P.: The inspherical Gergonne center of a tetrahedron. J. Geom. Graph. 8, 23–32 (2004)
Hajja M., Walker P.: Equifaciality of tetrahedra whose incenter and Fermat-Torricelli center coincide. J. Geom. Graph. 9, 37–41 (2005)
Heath, T. L.: Euclid–The Thirteen Books of the Elements, vol. 1, 2nd edn. Dover Publications, Inc., New York (1956)
Honsberger, R.,: Mathematical Gems II, Dolciani Math. Expositions No. 2, Math. Assoc. America. Washington (1976)
Honsberger, R.: In Pólya’s Footsteps, Miscellaneous Problems and Essays. Dolciani Math. Expositions No. 19, Math. Assoc. America, Washington (1997)
Liu, A.: Hungarian Problem Book III. Anneli Lax New Mathematical Library No. 42, Math. Assoc. America, Washington(2001)
Lob H.: The orthocentric simplex in space of three and higher dimensions. Math. Gaz. 19, 102–108 (1935)
Prasolov, V. V., Tikhomirov, V. M.: Geometry, Translations of Mathematical Monographs, vol. 200, AMS, Providence (2001)
Rosenbaum, J.: Problem 3644, Amer. Math. Mon. 40, 561–562 (1933); solution, ibid, 42, 51–53 (1935).
Stahl S.: Geometry, From Euclid to Knots. Prentice Hall, New Jersey (2003)
Venema G.A.: Geometry. Prentice Hall, New Jersey (2006)
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This work is supported by a research grant from Yarmouk University.
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Hajja, M. The pons asinorum and related theorems for tetrahedra. Beitr Algebra Geom 53, 487–505 (2012). https://doi.org/10.1007/s13366-011-0056-4
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DOI: https://doi.org/10.1007/s13366-011-0056-4
Keywords
- Bridge of asses
- Circumscriptible tetrahedra
- Content of a polyhedral angle
- Distinguished tetrahedra
- Equifacial tetrahedra
- Isodynamic tetrahedra
- Isogonic tetrahedra
- Isometry
- Orthocentric tetrahedra
- Polar sine
- Polyhedral angle
- Pons asinorum
- Rectangular simplex
- Sine of a polyhedral angle