  It is well-known that pythagorean triples can be represented by points of the
unit circle with rational coordinates. These points form an abelian group, and
we describe its structure. This structural description yields, almost
immediately, an enumeration of the normalized pythagorean triples with a given
hypotenuse, and also to an effective method for producing all such triples.
This effective method seems to be new.
  This paper is intended for the general mathematical audience, including
undergraduate mathematics students, and therefore it contains plenty of
background material, some history and several examples and exercises.
