Spherical Heron triangles and elliptic curves. (English) Zbl 1534.11086
Problem D21 in R. K. Guy’s book [Am. Math. Mon. 102, No. 9, 771–781 (1995; Zbl 0847.11031)] is an example of the “rationality” of triangles, and it asks whether there is a triangle such that the area, the sides, and the medians have rational values. Introduced in [R. Hartshorne and R. Van Luijk, Math. Intell. 30, No. 4, 4–10 (2008; Zbl 1227.14039)] is the idea of the rationality of hyperbolic triangles, and two of the authors of the paper under review studied in [J. Laflamme and M. Lalín, J. Number Theory 222, 48–74 (2021; Zbl 1475.11114)] various rationality problems of hyperbolic triangles. The paper under review concerns the rationality of “proper” triangles on the unit sphere, which are called spherical (proper) triangles.
Let \(x\) represent a side length, an angle, or the area of a spherical triangle. If \(\{\cos(x),\sin(x)\}\subset\mathbb Q\), then the side, the angle, or the area is called rational. Via the parametrization \(\cos(x)=(1-t^2)/(1+t^2)\) and \(\sin(x)=2t/(1+t^2)\), we have \(t=\sin(x)/(1 +\cos(x))\), and \(t\) may be called the side, angle, or area parameter. By the Gauss-Bonnet theorem, if the angles are rational, then the area is rational, and if all sides and angles are rational, it is called a spherical Heron triangle. Firstly, the authors prove the existence of various spherical Heron triangles: For all but finitely many pairs of rational sides, there are infinitely many spherical Heron triangles with the two rational sides.
Secondly, the authors prove the following version of the congruent number problem for spherical Heron right triangles: Let \(m\) be a rational area parameter \(\not\in\{0,-1\}\). If \(m\ne 1\), then there are infinitely many spherical Heron right triangles with area \(A\) such that \(m=\sin(A)/(1+\cos(A))\). If \(m=1\), then there is only one such spherical Heron right triangle. This is proved via making a connection to the elliptic curve given by \(y^2=x(x - 2m(m^2+1))(x - 4 m (m^2+1))\).
Considered also in the paper are an affirmative answer to the Problem D21 for spherical Heron triangles, and examples of the rationality of various cevians of spherical Heron triangles such as medians, angle/area-bisectors, and heights.
Let \(x\) represent a side length, an angle, or the area of a spherical triangle. If \(\{\cos(x),\sin(x)\}\subset\mathbb Q\), then the side, the angle, or the area is called rational. Via the parametrization \(\cos(x)=(1-t^2)/(1+t^2)\) and \(\sin(x)=2t/(1+t^2)\), we have \(t=\sin(x)/(1 +\cos(x))\), and \(t\) may be called the side, angle, or area parameter. By the Gauss-Bonnet theorem, if the angles are rational, then the area is rational, and if all sides and angles are rational, it is called a spherical Heron triangle. Firstly, the authors prove the existence of various spherical Heron triangles: For all but finitely many pairs of rational sides, there are infinitely many spherical Heron triangles with the two rational sides.
Secondly, the authors prove the following version of the congruent number problem for spherical Heron right triangles: Let \(m\) be a rational area parameter \(\not\in\{0,-1\}\). If \(m\ne 1\), then there are infinitely many spherical Heron right triangles with area \(A\) such that \(m=\sin(A)/(1+\cos(A))\). If \(m=1\), then there is only one such spherical Heron right triangle. This is proved via making a connection to the elliptic curve given by \(y^2=x(x - 2m(m^2+1))(x - 4 m (m^2+1))\).
Considered also in the paper are an affirmative answer to the Problem D21 for spherical Heron triangles, and examples of the rationality of various cevians of spherical Heron triangles such as medians, angle/area-bisectors, and heights.
Reviewer: Sungkon Chang (Savannah)
MSC:
| 11G05 | Elliptic curves over global fields |
| 14J27 | Elliptic surfaces, elliptic or Calabi-Yau fibrations |
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 14H52 | Elliptic curves |
| 11D25 | Cubic and quartic Diophantine equations |
Software:
QCIReferences:
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