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arXiv · 2609.38544

Euclidean Triangles Associated with Diophantine $D(4)$-triples

Marija Bliznac Trebješanin·Jelena Pleština
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Abstract

In this paper we study Euclidean triangles associated with Diophantine $D(4)$-triples. For a $D(4)$-triple $\{a,b,c\}$, $a b-a$. 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.

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Marija Bliznac Trebješanin, Jelena Pleština. 2026-09-29. Euclidean Triangles Associated with Diophantine $D(4)$-triples. https://arxiv.org/abs/2609.38544

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