The Seventeen Elements of Pythagorean Triangles

TL;DR

This paper provides a comprehensive analysis of the seventeen key elements of Pythagorean triangles, detailing conditions for their rationality, irrationality, or integrality, and offering precise criteria for each case.

Contribution

It systematically characterizes the rationality and integrality conditions for all seventeen elements of Pythagorean triangles, filling gaps in the geometric number theory literature.

Findings

Explicit conditions for the integrality of each element.

Criteria for the rationality of the elements.

Conditions under which elements are irrational.

Abstract

This is an exhaustive study of the seventeen elements of Pythagorean triangles, from the point of view of when such an element is an irrational number, a rational number, or an integer. For each of these 17 elements,precice conditions for their integrality,rationality, or irrationality; are given. These 17 elements are:The radius of the incircle, the radius of the circumscribed circle,the three radii of the three exterior circles tangential to the three lines containing the triangles'sides, the lengths of the three heights, the lenghts of the three internal angle bisectors, the lengthe of the three external angle bisectors, and the lengths of the three medians.