Certain Properties of Pythagorean Triangles involving the interior diameter, and the exterior diameters

TL;DR

This paper investigates Pythagorean triangles with integer diameters of characteristic circles, focusing on cases where a leg or diameter is a perfect square, and provides complete parametric descriptions of the feasible cases.

Contribution

It identifies which combinations of perfect square properties are possible in Pythagorean triangles and offers explicit parametric formulas for these cases.

Findings

Six out of eight possible square-combinations are realizable.

Complete parametric descriptions for each feasible combination.

Characterization of integer diameters in Pythagorean triangles.

Abstract

There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is Pythagorean, the four diameters are integers. Consider a Pythagorean triangle with the property that one leglength is a perfect(or integer)square, and with one of the four diameters also a integer square.Of the eight resulting combinations, we prove that only six are possible or can occur. We then completely parametrically describe the six families; each corresponding to one of the six combinations.