The circle as generator of Pythagorean triangles
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.
scientific article; zbMATH DE number 7855522