The circle as generator of Pythagorean triangles
type: Item
instance of: scholarly article
title: The circle as generator of Pythagorean triangles
zbMATH Open document ID: 1547.11043
author: M. O'Loughlin
published in: Elemente der Mathematik
publication date: 2024-05-29T00:00:00Z
review text: A right-angled triangle whose side lengths are natural numbers is termed a \textit{Pythagorean triangle}. The central idea of the author of paper under review is that circles generate Pythagorean triangles and they do so in a systematic way. Every circle of integral radius generates a set number of such triangles and every Pythagorean triangle is generated by some circle. Thus, circles provide a simple way of enumerating Pythagorean triangles. The main result of the author is the following.\N\NTheorem. For \( n,i,j\in \mathbb{N}\), where \( d =2n\) is the length of the diameter of a circle and \( i<j \), the triangle with side lengths \( (d+i,d+j,d+i+j) \), with \( ij=\frac{d^2}{2}=2r^2 \) is a right-angled triangle whose incircle has diameter length \( d \). Furthermore, every Pythagorean triangle \( (a,b,c) \) is of this form.
reviewed by: Mahadi Ddamulira
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 51M04
zbMATH DE Number: 7855522
zbMATH Keywords: circle
zbMATH Keywords: Pythagorean triangle
MaRDI profile type: Publication
cites work: Tabulating All Pythagorean Triples
cites work: Neither Sherlock Holmes nor Babylon: A reassessment of Plimpton 322
DOI: 10.4171/EM/505
description: scientific article; zbMATH DE number 7855522