Skip to main content
SpringerLink
Log in

Some Inequalities for Triangle: Old and New Results

  • Chapter
Analytic and Geometric Inequalities and Applications

Part of the book series: Mathematics and Its Applications ((MAIA,volume 478))

Abstract

In this paper a brief review of known results from the area of inequalities for triangle is given, the classification of these results is done, and for ineqalities chosen in this way, their hierarchy, severity and some of their generalizations are pointed out. Also, a three new inequalities for elements of triangle are proved.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Andersson, Problem E 1779, Amer. Math. Monthly 59 (1952), 41.

    Article  MathSciNet  Google Scholar 

  2. A. Bager and H. Frischknecht, Aufgabe 716, Elem. Math. 27 (1972), 68 and 28 (1973), 75.

    Google Scholar 

  3. A. Bager and O. Reuter, Aufgabe 688, Elem. Math. 29 (1974), 18–19.

    Google Scholar 

  4. A. Bager and O. Reuter, Aufgabe 690, Elem. Math. 29 (1974), 46–47.

    Google Scholar 

  5. F. Balitrand, Interméd. Math. 23 (1916), 86–87.

    Google Scholar 

  6. B. M. Barbalatt, Gaz. Mat. 30 (1924/25), 380–381.

    Google Scholar 

  7. I. Bénézech, Problem 412, J. Math. Elém. Paris (4) 1 (1892), 234.

    Google Scholar 

  8. W. J. Blundon, Inequalities Associated with the Triangle, Canad. Math. Bull. 8 (1965), 615–626.

    Article  MathSciNet  MATH  Google Scholar 

  9. W. J. Blundon, Problem E 1925, Amer. Math. Monthly 73 (1966), 1016.

    Article  MathSciNet  Google Scholar 

  10. W. J. Blundon, Problem E 1935, Amer. Math. Monthly 73 (1966), 1122.

    Article  MathSciNet  Google Scholar 

  11. O. Bottema, Euclides 30 (1954/55), 114–116.

    MathSciNet  Google Scholar 

  12. O. Bottema, R. Ž. Djordjević, R. R. Janić, D. S. Mitrinović and P. M. Vasić, Geometric Inequalities, Wolters-Noordhoff Publishing, Groningen, 1969.

    MATH  Google Scholar 

  13. O. Bottema and G. R. Veldkamp, Problem 364, Nieuw Arch. Wisk. 22 (1974), 79 and 22 (1974), 266-267.

    Google Scholar 

  14. L. Carlitz and F. Leuenberger, Problem E 1454, Amer. Math. Monthly 68 (1961), 177 and 68 (1961), 805-806.

    Article  MathSciNet  Google Scholar 

  15. L. Carlitz and S. Philipp, Problem E 1828, Amer. Math. Monthly 70 (1963), 891 and 71 (1964), 681.

    Google Scholar 

  16. L. Carlitz, Problem E 1628, Amer. Math. Monthly 70 (1963), 891 and 71 (1964), 687.

    Google Scholar 

  17. J. M. Child, Math. Gaz. 23 (1939), 138–143.

    Article  Google Scholar 

  18. M. Collins and A. B. Evans, Problem 2837, Ed. Times 13 (1870), 30–31.

    Google Scholar 

  19. G. Colombier and T. Doucet, Problem 1051, Nouv. Ann. Math. 31 (1872), 467.

    Google Scholar 

  20. V. Cristescu, Gaz. Mat. 26 (1920/21), 88–89.

    Google Scholar 

  21. T. R. Curry, Problem E 1861, Amer. Math. Monthly 73 (1966), 199.

    MathSciNet  Google Scholar 

  22. T. R. Curry, Problem E 1644, Amer. Math. Monthly 70 (1963), 1099 and 71 (1964), 915-916.

    Article  MathSciNet  Google Scholar 

  23. R. Z. Djordjević, Some Inequalities for Triangle, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No 247-No 273 (1969), 35–39.

    Google Scholar 

  24. R. Z. Djordjević, Some Inequalities for the Elements of a Triangle, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No 274-No 301 (1969), 124–126.

    Google Scholar 

  25. R. Z. Djordjević, Some Inequalities for Elements of a Triangle, Facta Universitatis (Nis) Ser. Math. Inform. 8 (1993), 73–75.

    MATH  Google Scholar 

  26. N. A. Edwards, Problem 1273, Nouv. Ann. Math. 37 (1878), 475.

    Google Scholar 

  27. A. Emerich, Problem 10656, Math. Questions 54 (1891), 100.

    Google Scholar 

  28. L. Euleri, Novi commentarii academiae scientarium Petropolitanae 11 (1765), 103–123.

    Google Scholar 

  29. L. Euleri, Opera omnia, I 26, 1953, pp. 139–157.

    Google Scholar 

  30. P. Finsler and H. Hadwigwr, Comment. Math. Helv. 10 (1937/38), 316–326.

    Article  MathSciNet  Google Scholar 

  31. Ju. I. Gerasimov, Mat. v škole No 3 (1964), 75.

    Google Scholar 

  32. J. C. H. Gerretsen, Ongelijkheden in de Driehoek, Nieuw Tijdschr. Wisk. 41 (1953), 1–7.

    MathSciNet  Google Scholar 

  33. F. Goldner, Problem 69, Elem. Math. 4 (1949), 120.

    Google Scholar 

  34. V. O. Gordon, Mat. v škole No 5 (1966), 80.

    Google Scholar 

  35. E. G. Gotman, Mat. v škole No 1 (1965), 76.

    Google Scholar 

  36. E. G. Gotman, Mat. v škole No 5 (1966), 76.

    Google Scholar 

  37. S. G. Guba, Mat. v škole No 5 (1965), 69.

    Google Scholar 

  38. S. Guba, Problem 317, Mat. v škole No 6 (1966), 67.

    Google Scholar 

  39. H. W. Guggenheimer, Problem E 1724, Amer. Math. Monthly 71yr 1964, 911 and 72 (1965), 791–793.

    Google Scholar 

  40. H. W. Guggenheimer, Plane Geometry and its Groups, San Francisco-Cambridge-London-Amsterdam 1967.

    Google Scholar 

  41. H. Hadwiger, Jber. Deutsch. Math.-Verein. 49 (1939), 35–39.

    MATH  Google Scholar 

  42. T. Hayashi, Tôhoku Science Rep. 11 (1922), 115–121.

    Google Scholar 

  43. R. R. Janić, On a Geometric Inequality of D. F. Barrow, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No 181-No 196 (1967), 75–76.

    Google Scholar 

  44. J. Karamata, Problem 119, Glasnik Mat.-Fiz. Astronom. 3 (1948), 223.

    Google Scholar 

  45. M. S. Klamkin, Math. Teacher 60 (1967), 323–328.

    Google Scholar 

  46. O. Kooi, Simon Stevin 32 (1958), 97–101.

    MathSciNet  MATH  Google Scholar 

  47. R. Kooistra, Nieuw Tijdschr. Wisk. 45 (1957/58), 108–115.

    MathSciNet  Google Scholar 

  48. T. Kubota, Tôhoku Math. J. 25 (1925), 122–126.

    MATH  Google Scholar 

  49. M. N. Kritikos, Actes du Congrès interbalkanique de mathématiciens, Athènes, 1934, pp. 157-158.

    Google Scholar 

  50. A. Laisant, Géométrie du triangle, Paris, 1896.

    Google Scholar 

  51. E. Lemoine, Problem 578, J. Math. Elem. (4) 3 (1894), 263.

    Google Scholar 

  52. F. Leuenberger, Einige Dreiecksungleichungen, Elem. Math. 13 (1958), 121–126.

    MathSciNet  MATH  Google Scholar 

  53. F. Leuenberger, Elem. Math. 15 (1960), 77–79.

    MathSciNet  MATH  Google Scholar 

  54. F. Leuenberger, Gegensätzliches Verhalten der arithmetischen und geometrischen Mittel, Elem. Math. 16 (1961), 127–129.

    Google Scholar 

  55. F. Leuenberger, Elem. Math. 18 (1963), 35–36.

    MATH  Google Scholar 

  56. F. Leuenberger and L. Carlitz, Problem E 1481, Amer. Math. Monthly 68 (1961), 803 and 69 (1962), 312.

    Article  MathSciNet  Google Scholar 

  57. A. Lupaş, Problem 441, Mat. Vesnik 2 (15)(30) (1978), 293.

    Google Scholar 

  58. A. Makowski, Elem. Math. 16 (1961), 134.

    Google Scholar 

  59. A. Makowski and J. Schopp, Problem E 1675, Amer. Math. Monthly 72 (1965), 187–188.

    Article  MathSciNet  Google Scholar 

  60. M. Marčev, Neravenstva meždu perimetr’a i radiusite na vpisanata i opisanata okr’uznost na tri’g’lnika i njakoi sledstvija ot tjah’, Ob. po matematika (Sofija), No 6 (1976), 3–7.

    Google Scholar 

  61. D. Marković, Bull. Soc. Math. Phys. Serbie (3–4) 4 (1952), 71.

    Google Scholar 

  62. C. N. Mills and O. Dunkel, Problem 3207, Amer. Math. Monthly 34 (1927), 382–384.

    Article  MathSciNet  Google Scholar 

  63. D. S. Mitrinović, J. E. Pečarić and V. Volonec, Recent Advances in Geometric Inequalities, Springer Science+Business Media Dordrecht, Dordrecht-Boston-London, 1989.

    MATH  Google Scholar 

  64. S. Nakajima, Some Inequalities between the Fundamental Quantities of the Triangle, Tôhoku Math. J. 25 (1925), 115–121.

    MATH  Google Scholar 

  65. J. Neuberg, Ed. Times, News Ser. 9 (1906), 51–52.

    Google Scholar 

  66. A. Padoa, Period. Mat. 5(4) (1925), 80–85.

    MATH  Google Scholar 

  67. A. Padoa, Period. Mat. 6(4) (1926), 38–40.

    MATH  Google Scholar 

  68. A. Pantazi, Gaz. Mat. 23 (1917/18), 144.

    Google Scholar 

  69. I. Pasche, Problem 165, Mat. Vesrtik 10 (25) (1973), 97.

    Google Scholar 

  70. M. Petrović, Sur quelques fonctions des côtés et des angles d’une triangle, Ens. Math. 18 (1916), 153–163.

    Google Scholar 

  71. C. C. Popovici, Gaz. Mat. 31 (1925), 132.

    MATH  Google Scholar 

  72. Ramus and é. Rouché, Question 233, Nouv. Ann. Math. 10 (1851), 353-355.

    Google Scholar 

  73. F. Ryžkov, Mat. v škole No 1 (1964), 80 and No 4 (1964), 73.

    Google Scholar 

  74. L. A. Santalo, Math. Notae 3(2) (1943), 65–73.

    MathSciNet  MATH  Google Scholar 

  75. K. Schuler, Praxis der Math. 9 (1967), 344.

    Google Scholar 

  76. C. Sebastiano, Boll. Math. (Firenze) (1938), 59.

    Google Scholar 

  77. Z. A. Skopec, Mat. v škole No 3 (1963), 89.

    Google Scholar 

  78. J. Steinig, Elem. Math. 18 (1963), 127–131.

    MathSciNet  MATH  Google Scholar 

  79. G. Tsintsifas and M. S. Klamkin, Problem 816, Crux Math. 2 (1983), 46 and 10 (1984), 157.

    Google Scholar 

  80. T. Varopoulos, Bull. Soc. Grèce 15, (1934), 17.

    MATH  Google Scholar 

  81. R. Weitzenböck, Math. Z. 5 (1919), 137–146.

    Article  MathSciNet  MATH  Google Scholar 

  82. K. P. Wilkins, Amer. Math. Monthly 44 (1937), 579–583.

    Article  MathSciNet  Google Scholar 

  83. S. Zetel’, Mat. v škole No 3 (1965), 66–69.

    Google Scholar 

  84. V. G. Zvezdin, Mat. v škole (1968), 65.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Electronic Engineering, Department of Mathematics, P. O. Box 73, 18000, Niš, Yugoslavia

    Radosav Ž. Djordjević

Authors
  1. Radosav Ž. Djordjević
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens, Greece

    Themistocles M. Rassias

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    Hari M. Srivastava

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Djordjević, R.Ž. (1999). Some Inequalities for Triangle: Old and New Results. In: Rassias, T.M., Srivastava, H.M. (eds) Analytic and Geometric Inequalities and Applications. Mathematics and Its Applications, vol 478. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4577-0_6

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-4577-0_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5938-1

  • Online ISBN: 978-94-011-4577-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation