Skip to content
DraconiticResearch
Search Sources API
194,164 studies31 sourcesUpdated 36 min ago
PreprintCC BY-NC-ND 4.0arXiv

Euclidean Triangles Associated with Diophantine D(4)-triples

  • Marija Bliznac Trebješanin
  • Jelena Pleština
Open at sourcePDFDOICite

Abstract

In this paper we study Euclidean triangles associated with Diophantine D(4)-triples. For a D(4)-triple {a,b,c}, a<b<c, we prove that the lengths r=√(ab+4), s=√(ac+4) and t=√(bc+4) associated with this D(4)-triple form a non-degenerate triangle if and only if the triple is regular and r>b−a. For a fixed shortest side r, we describe all triangles associated with D(4)-triples in terms of the factorisations of r2−4, give a divisor criterion for their number, and determine their angle type. We also prove that two similar triangles associated with D(4)-triples are necessarily congruent. Finally, for the family of D(4)-triples {r−2,r+2,4r}, we obtain a Pell-equation characterisation of the Heronian triangles associated with these D(4)-triples.

Related

  1. Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres
    Mahdi Hormozi, Jakub Niksiński, Błażej Wróbel
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  2. A new proof of two-color partition regularity of Pythagorean triples
    James Leng
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  3. A note on realizing triples of slice, concordance and Seifert genera
    Boning Wang
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  4. Numerical index one characterises commutative JB*-triples
    Lei Li, Siyu Liu, Antonio M. Peralta
    arXivPreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsarXiv
  5. Surgery Exact Triangles in Involutive Heegaard Floer Homology
    Kristen Hendricks, Jennifer Hom, Matthew Stoffregen +1
    Memoirs of the American Mathematical SocietyPeer-reviewedLicence not statedFull text31 citedGeometry & topologyOpenAlex

Cite

BibTeXRIS
BibTeX
@misc{trebjesanin2026euclidean,
  title = {{Euclidean Triangles Associated with Diophantine $D(4)$-triples}},
  author = {Marija Bliznac Trebješanin and Jelena Pleština},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38544},
  eprint = {2609.38544},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38544},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Euclidean Triangles Associated with Diophantine D(4)-triples
AU  - Marija Bliznac Trebješanin
AU  - Jelena Pleština
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.38544
UR  - https://arxiv.org/abs/2609.38544
AB  - In this paper we study Euclidean triangles associated with Diophantine D(4)-triples. For a D(4)-triple {a,b,c}, a<b<c, we prove that the lengths r=√(ab+4), s=√(ac+4) and t=√(bc+4) associated with this D(4)-triple form a non-degenerate triangle if and only if the triple is regular and r>b−a. For a fixed shortest side r, we describe all triangles associated with D(4)-triples in terms of the factorisations of r^2−4, give a divisor criterion for their number, and determine their angle type. We also prove that two similar triangles associated with D(4)-triples are necessarily congruent. Finally, for the family of D(4)-triples {r−2,r+2,4r}, we obtain a Pell-equation characterisation of the Heronian triangles associated with these D(4)-triples.
ER  -