Advertisement

Journal of Geometry

, 110:36 | Cite as

An algebraic characterization of properly congruent-like quadrilaterals

  • Giuseppina Anatriello
  • Francesco Laudano
  • Giovanni VincenziEmail author
Article
  • 10 Downloads

Abstract

In this paper, we will provide a characterization of pairs of non-congruent quadrilaterals for which all elements are pairwise congruent (‘properly congruent-like quadrilaterals’). As a consequence of this main result, we demonstrate a method to establish, given a generic quadrilateral, whether some quadrilaterals that are properly congruent-like to it exist and, if so, how to determine the values of their elements. In particular, this approach allows us to provide examples of quadrilaterals that are not congruent-like to any other quadrilateral and to show constructive examples of pairs of properly congruent-like quadrilaterals.

Keywords

Quadrilaterals Congruence theorems Congruent-like quadrilaterals Properly congruent-like quadrilaterals 

Mathematics Subject Classification

51M04 51M05 97G40 

Notes

Compliance with ethical standards

Conflict of interest

No potential conflict of interest was reported by the authors.

References

  1. 1.
    Laudano, F., Vincenzi, G.: Congruence theorems for quadrilaterals. J. Geom. Graphics 21(1), 45–59 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anatriello, G., Laudano, F., Vincenzi, G.: Pairs of congruent-like quadrilaterals that are not congruent. Forum Geom. 18, 381–400 (2018)MathSciNetGoogle Scholar
  3. 3.
    Moise, E.: Elementary Geometry from an Advanced Standpoint, 3rd edn. Addison-Wesley Publishing Company, Reading (1990)zbMATHGoogle Scholar
  4. 4.
    Harvey, M.: Geometry Illuminated. An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry. MAA TextbooksMathematical Association of America, Washington (2015)zbMATHGoogle Scholar
  5. 5.
    Johnson, R.A.: Advanced Euclidean Geometry, p. 82. Dover Publishing Company, Mineola (2007)Google Scholar
  6. 6.
    Schwarz, D., Smith, G.C.: On the three diagonals of a cyclic quadrilateral. J. Geom. 105(2), 307–312 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Laudano, F., Vincenzi, G.: Continue quadrilaterals. Math. Commun. 24, 133–146 (2019)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Josefsson, M.: Characterizations of orthodiagonal quadrilaterals. Forum Geom. 12, 13–25 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Josefsson, M.: Properties of equidiagonal quadrilaterals. Forum Geom. 14, 129–144 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Lee, J.M.: Axiomatic Geometry. Pure and Applied Undergraduate TextsAmerican Mathematical Society, Providence (2013)zbMATHGoogle Scholar
  11. 11.
    Peter, T.: Maximizing the area of a quadrilateral. College Math. J. 34(4), 315–316 (2003)CrossRefGoogle Scholar
  12. 12.
    Pierro, F., Vincenzi, G.: On a conjecture referring to orthic quadrilaterals. Beitr. Algebra Geom. 57, 441–451 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Usiskin, Z., Griffin, J., Witonsky, D., Willmore, E.: The Classification of Quadrilaterals: A Study of Definition. Information Age Pubblishing, Charlotte (2008)Google Scholar
  14. 14.
    Martini, H.: Recent results in elementary geometry. Part II. In: Behara, M., Fritsch, R., Lintz, R.G. (eds) Proceedings of the 2nd Gauss Symposium. Conference A: Mathematics and Theoretical Physics. (Munich, 1993), Sympos. Gaussiana, Gruyter, Berlin, pp. 419–443 (1995)Google Scholar
  15. 15.
    Syropoulos, A.: Mathematics of Multisets. Multiset Processing. Mathematical, Computer Science, and Molecular Computing Points of View. Lecture Notes in Computer Science, vol. 2235. Springer, Berlin (2001) ISBN: 3-540-43063-668-06 (68Q05)Google Scholar
  16. 16.
    Calcut, Jack S.: Grade school triangles. Am. Math. Mon. 117, 673–685 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di di ArchitetturaUniversità di Napoli “Federico II”NapoliItaly
  2. 2.Liceo Scientifico RomitaCampobassoItaly
  3. 3.Dipartimento di MatematicaUniversità di SalernoFiscianoItaly

Personalised recommendations