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On rational triangles via algebraic curves. (English) Zbl 1430.11087

Summary: A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between each family and the set of rational points of an algebraic curve. These algebraic curves are: a curve of genus 0, an elliptic curve and a genus 3 curve. We study the set of rational points on each of these curves and explicitly describe some of its rational points.

MSC:

11G05 Elliptic curves over global fields
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
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