Journal of Geometry

, 110:36

# An algebraic characterization of properly congruent-like quadrilaterals

• Giuseppina Anatriello
• Francesco Laudano
• Giovanni Vincenzi
Article

## 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.

## Mathematics Subject Classification

51M04 51M05 97G40

## Notes

### 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)
2. 2.
Anatriello, G., Laudano, F., Vincenzi, G.: Pairs of congruent-like quadrilaterals that are not congruent. Forum Geom. 18, 381–400 (2018)
3. 3.
4. 4.
Harvey, M.: Geometry Illuminated. An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry. MAA TextbooksMathematical Association of America, Washington (2015)
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)
7. 7.
Laudano, F., Vincenzi, G.: Continue quadrilaterals. Math. Commun. 24, 133–146 (2019)
8. 8.
Josefsson, M.: Characterizations of orthodiagonal quadrilaterals. Forum Geom. 12, 13–25 (2012)
9. 9.
Josefsson, M.: Properties of equidiagonal quadrilaterals. Forum Geom. 14, 129–144 (2014)
10. 10.
Lee, J.M.: Axiomatic Geometry. Pure and Applied Undergraduate TextsAmerican Mathematical Society, Providence (2013)
11. 11.
Peter, T.: Maximizing the area of a quadrilateral. College Math. J. 34(4), 315–316 (2003)
12. 12.
Pierro, F., Vincenzi, G.: On a conjecture referring to orthic quadrilaterals. Beitr. Algebra Geom. 57, 441–451 (2016)
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)