Li, Yangcheng; Zhang, Yong \(\theta\)-triangle and \(\omega\)-parallelogram pairs with areas and perimeters in certain proportions. (English) Zbl 1440.51006 Rocky Mt. J. Math. 50, No. 3, 1059-1071 (2020). Summary: By the theory of elliptic curves, we show that given a convex angle \(\theta\), there exist, except for finitely many exceptions, infinitely many pairs of rational \(\theta\)-triangle and \(\omega\)-parallelogram with areas and perimeters in fixed proportions \((\alpha, \beta)\) respectively, satisfying that \(\sin \omega\) is a previously fixed rational multiple of \(\sin \theta\), where \(\alpha\) and \(\beta\) are positive rational numbers. Cited in 3 Documents MSC: 51M25 Length, area and volume in real or complex geometry 11D25 Cubic and quartic Diophantine equations 11G05 Elliptic curves over global fields 51M05 Euclidean geometries (general) and generalizations Keywords:\(\theta\)-triangle; \(\omega\)-parallelogram; area; perimeter; elliptic curve PDFBibTeX XMLCite \textit{Y. Li} and \textit{Y. Zhang}, Rocky Mt. J. Math. 50, No. 3, 1059--1071 (2020; Zbl 1440.51006) Full Text: DOI Euclid References: [1] M. Aassila, “Some results on Heron triangles”, Elem. Math. 56:4 (2001), 143-146. · Zbl 1119.11307 [2] M. E. Bradley, “Isosceles triangles with the same perimeter and area”, Math. Teacher 73:4 (1980), 264-266. [3] A. Bremner, “On Heron triangles”, Ann. Math. Inform. 33 (2006), 15-21. · Zbl 1135.11315 [4] A. Bremner and R. K. Guy, “Triangle-rectangle pairs with a common area and a common perimeter”, Int. J. Number Theory 2:2 (2006), 217-223. · Zbl 1113.11032 [5] S. Chern, “Integral right triangle and rhombus pairs with a common area and a common perimeter”, Forum Geom. 16 (2016), 25-27. · Zbl 1335.51021 [6] A. Choudhry, “Rational triangles with the same perimeter and the same area”, Hardy-Ramanujan J. 30 (2007), 19-30. [7] A. Choudhry, “Brahmagupta quadrilaterals with equal perimeters and equal areas”, Int. J. Number Theory 16:3 (2020), 523-535. · Zbl 1460.11037 [8] P. Das, A. Juyal, and D. Moody, “Integral isosceles triangle-parallelogram and Heron triangle-rhombus pairs with a common area and common perimeter”, J. Number Theory 180 (2017), 208-218. · Zbl 1421.11025 [9] L. E. Dickson, History of the theory of numbers, II: Diophantine analysis, Chelsea, 1966. [10] R. K. Guy, “My favorite elliptic curve: a tale of two types of triangles”, Amer. Math. Monthly 102:9 (1995), 771-781. · Zbl 0847.11031 [11] Y. Hirakawa and H. Matsumura, “A unique pair of triangles”, J. Number Theory 194 (2019), 297-302. · Zbl 1400.14069 [12] A.-V. Kramer and F. Luca, “Some remarks on Heron triangles”, Acta Acad. Paedagog. Agriensis Sect. Math. \((\) N.S.\() 27 (2000), 25-38\). · Zbl 1062.11019 [13] M. Lalín and X. Ma, “\( \theta \)-triangle and \(\omega \)-parallelogram pairs with common area and common perimeter”, J. Number Theory 202 (2019), 1-26. · Zbl 1448.11106 [14] Y. C. Li and Y. Zhang, “Rational triangles pairs and cyclic quadrilaterals pairs with areas and perimeters in certain proportions”, submitted. [15] D. R. Lichtenberg, “More about triangles with the same area and the same perimeter”, Math. Teacher 67:7 (1974), 659-660. [16] R. van Luijk, “An elliptic \(K3\) surface associated to Heron triangles”, J. Number Theory 123:1 (2007), 92-119. · Zbl 1160.14029 [17] I. Newton, The mathematical papers of Isaac Newton, IV: 1674-1684, Cambridge Univ. Press, 1971. [18] R. W. Prielipp, “Are triangles that have the same area and perimeter congruent?”, Math. Teacher 67:2 (1974), 157-159. [19] K. R. S. Sastry, “Heron triangles: a Gergonne-cevian-and-median perspective”, Forum Geom. 1 (2001), 17-24. · Zbl 0986.51030 [20] J. H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics 106, Springer, 2009. · Zbl 1194.11005 [21] J. H. Silverman and J. Tate, Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer, 1992. · Zbl 0752.14034 [22] T. Skolem, Diophantische Gleichungen, vol. 5, Ergeb. Math. Grenzgeb. 4, Springer, 1938. [23] A. Wares, “Mutually non-congruent triangles with the same perimeter and the same area”, Internat. J. Math. Ed. Sci. Tech. 33:5 (2002), 788-791. [24] P. Yiu, “Isosceles triangles equal in perimeter and area”, Missouri J. Math. Sci. 10:2 (1998), 106-111. · Zbl 1097.51512 [25] Y. Zhang, “Right triangle and parallelogram pairs with a common area and a common perimeter”, J. Number Theory 164 (2016), 179-190. · Zbl 1391.11063 [26] Y. Zhang and J. Peng, “Heron triangle and rhombus pairs with a common area and a common perimeter”, Forum Geom. 17 (2017), 419-423. · Zbl 1381.51013 [27] Y. Zhang and A. S. Zargar, “Integral triangles and cyclic quadrilaterals pairs with a common area and a common perimeter”, preprint, 2019. To appear in Forum Geom. [28] Y. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.