Spherical Heron triangles and elliptic curves
Tinghao Huang1; Matilde Lalín2; Olivier Mila2
1 The Ohio State University (Columbus campus), Columbus, Ohio, 43210 The United States of America
2 Université de Montréal, Pavillon André-Aisenstadt, Département de mathématiques et de statistique, CP 6128, succ. Centre-ville, Montréal, Québec, H3C 3J7 Canada
Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 219-246.

We define spherical Heron triangles (spherical triangles with “rational” side-lengths and angles) and parametrize them via rational points of certain families of elliptic curves. We show that the congruent number problem has infinitely many solutions for most areas in the spherical setting and we find a spherical Heron triangle with rational medians. We also explore the question of spherical triangles with a single rational median or a single a rational area bisector (median splitting the triangle in half), and discuss various problems involving isosceles spherical triangles.

Nous définissons les triangles de Héron sphériques (triangles sphériques avec des mesures de côtés et des angles « rationnels ») et les paramétrons par des points rationnels en certaines familles de courbes elliptiques. Nous montrons que le problème des nombres congruents a une infinité de solutions pour la plupart des valeurs de l’aire dans le cas sphérique et nous trouvons un triangle de Héron sphérique avec des médianes rationnelles. Nous explorons également la question des triangles sphériques avec une seule médiane rationnelle ou une seule bissectrice d’aire rationnelle (c’est-à-dire, une médiane divisant le triangle en deux), et nous discutons de divers problèmes impliquant des triangles sphériques isocèles.

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DOI: 10.5802/jtnb.1243
Classification: 11G05, 14J27, 14J28, 14H52, 11D25
Keywords: spherical triangles, elliptic curves, elliptic surfaces
Tinghao Huang 1; Matilde Lalín 2; Olivier Mila 2

1 The Ohio State University (Columbus campus), Columbus, Ohio, 43210 The United States of America
2 Université de Montréal, Pavillon André-Aisenstadt, Département de mathématiques et de statistique, CP 6128, succ. Centre-ville, Montréal, Québec, H3C 3J7 Canada
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Tinghao Huang; Matilde Lalín; Olivier Mila. Spherical Heron triangles and elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 219-246. doi : 10.5802/jtnb.1243. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1243/

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