Rational triangles with equal area
type: Item
instance of: scholarly article
title: Rational triangles with equal area
zbMATH Open document ID: 0893.11009
author: David J. Rusin
published in: The New York Journal of Mathematics
publication date: 1998-04-21T00:00:00Z
full work available at URL: https://eudml.org/doc/119520
full work available at URL: 4-1.html
review text: The key result is that, given a rational triangle \((p,q,r)\) with area \(A\), there are infinitely many rational triangles with area \(A\) and one side equal to \(r\), unless \(p=q\) or one of two other conditions hold. This is obtained by studying the surface defined by \[ C=-(p+q+r)(p+q-r)(p-q+r)(-p+q+r) \] with \(C\) a negative real, which can be transformed into an elliptic curve whose group structure is analyzed. The author studies in detail triangles satisfying one of the exceptional conditions, and ends up with the result that infinitely many triangles can be constructed sharing a side and the area with a given non-equilateral triangle. Finally, he looks at when the elliptic curves are isomorphic. There are suggestions for further investigation.
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11D09
zbMATH DE Number: 1144094
zbMATH Keywords: Heron triangles
zbMATH Keywords: equal area
zbMATH Keywords: quadratic diophantine equations
zbMATH Keywords: quartic diophantine equations
zbMATH Keywords: cubic diophantine equations
zbMATH Keywords: rational triangles
zbMATH Keywords: elliptic curve
reviewed by: Edward J. Barbeau
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1144094