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\({\theta}\)-triangle and \({\omega}\)-parallelogram pairs with common area and common perimeter. (English) Zbl 1448.11106

Summary: We show that given a convex angle \({\theta}\), there exist, except for finitely many exceptions, infinitely many pairs of integral \({\theta}\)-triangle and \({\omega}\)-parallelogram with common area and common perimeter satisfying that \(\sin \omega\) is a previously fixed rational multiple of \(\sin \theta\). This is achieved by relating the problem to a family of elliptic curves. We also study the elliptic surfaces and the elliptic threefold that are obtained by varying one or two parameters.

MSC:

11G05 Elliptic curves over global fields
14H52 Elliptic curves
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
11D25 Cubic and quartic Diophantine equations
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