  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.
