In this Article, we derive a method for classifying all positive integer solutions of the equation x2 + ry2 = z2, where r is a given positive rational. In order to simplify notation, such a solution with x = a, y = b and z = c will be denoted by the ordered triple (a, b, c) and, in all that follows, the term solution will be taken to mean a positive integer solution of the above equation. The method employed will be similar to that used in [1] to find all integer triangles containing an angle whose cosine is known.