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Construction of Indecomposable Heronian Triangles
Rocky Mountain Journal of Mathematics, 1998 A Heronian triangle has integer sides and area. It is decomposable iff it is obtained by juxtaposing two right triangles or excising one from another. A triangle with rational sides has rational area iff the tangents \(t_1, t_2, t_3\) of its half angles are rationals. We have \(\sum t_it_j=1\); if \(t_i= n_i/d_i\) in lowest terms, the quantities \(g_i=
Paul Yiu
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ON LATTICE POINTS WHICH BECOME VERTICES OF HERONIAN TRIANGLES
Far East Journal of Mathematical Sciences, 2018 In this paper, we reveal that every Heronian triangle can be realized as a lattice triangle.
null Koichi Arimoto +1 more
exaly +4 more sources
On the Generation of Heronian Triangles [PDF]
Serdica Journal of Computing, 2008 We describe several algorithms for the generation of integer Heronian triangles with diameter at most n. Two of them have running time O(n^(2+ε)). We enumerate all integer Heronian triangles for n ≤ 600000 and apply the complete list on some related problems.
Sascha Kurz
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On quasi-Heronian equable triangles
Elemente der Mathematik, 2023 Triangles having integer area and side lengths are said to be \textit{Heronian}. Moreover, if their perimeter and area have the same value, they are called \textit{equable}. The authors study the set of equable triangles having side lengths \(b+\sqrt a\), \(b-\sqrt a\), \(c\), where \(a, b, c\in \mathbb N\) and \(c\) is a non-square, which they define ...
Christian Aebi
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Pseudo-Heronian triangles whose squares of the lengths of one or two sides are prime numbers
Lietuvos Matematikos Rinkinys, 2021 The authors introduced the concept of a pseudo-Heron triangle, such that squares of sides are integers, and the area is an integer multiplied by $2$. The article investigates the case of pseudo-Heron triangles such that the squares of the two sides of ...
Edmundas Mazėtis +1 more
doaj +4 more sources
Rational cuboids and Heron triangles II
Lietuvos Matematikos Rinkinys, 2019 We study the connection of Heronian triangles with the problem of the existence of rational cuboids. It is proved that the existence of a rational cuboid is equivalent to the existence of a rectangular tetrahedron, which all sides are rational and the ...
Edmundas Mazėtis +1 more
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A rational sine and cosine of the angles of a triangle
Lietuvos Matematikos Rinkinys, 2013 The article deals with triangles whose sines and cosines of the angles are the rational numbers. We prove that these triangles are similar to Pythagorean or Heronian triangles, and vice versa.
Edmundas Mazėtis +1 more
doaj +2 more sources
Construction of an infinite family of elliptic curves of 2-Selmer rank 1 from Heron triangles [PDF]
Research in Number Theory, 2022 Given any positive integer n , there exist triangles, called Heron triangles, with rational sides whose area is n . Assuming the finiteness of the Shafarevich–Tate group, we construct a family of infinitely many Heronian elliptic curves of rank 1 arising
Debopam Chakraborty, V. Ghale, A. Saikia
semanticscholar +1 more source
A rational cotangents of the angles of a triangle
Lietuvos Matematikos Rinkinys, 2014 We investigate how rational values cotangents angles of a triangle influence of its properties. The main result: cotangents of any two angles of a triangle are rational then, and only then, when a triangle similar to a triangle whose squares of sides are
Edmundas Mazėtis +1 more
doaj +1 more source
Serdica Journal of Computing, 2013 A set of n lattice points in the plane, no three on a line and no four on a circle, such that all pairwise distances and coordinates are integers is called an n-cluster (in R^2).
Sascha Kurz +3 more
semanticscholar +1 more source