In chapter V of volume 2 of Dickson's "History of the theory of numbers" (p.191-195), which collects results on "rational" triangles (triangles with integer side lengths), apear several results on the so called "Curtius's problem" (also called "the problem of three shooters"). This problem is concerned with establishing a rule for finding rational triangles which also have circumscribed circles with integer radius.
On p.195, appears the following remark:
C.F Gauss, whose attention had been called to Curtius's problem by Schumacher, stated that the sides of every triangle such that each side and the radius $r$ of the circumscribed circle are integers are of the form: $$4abfg(a^2+b^2), \pm4ab(f+g)(a^2f-b^2g), 4ab(a^2f^2+b^2g^2)$$ where $a,b,f,g$ are positive integers, while $r = (a^2+b^2)(a^2f^2+b^2g^2)$... Many writers derived Gauss's formula.
I checked in Gauss's werke, and those formulas are given in a letter to Schumacher from 1847, which appears in volume 12 under the title "solution to arithmetical task". What i want to know is how to put this isolated result in modern mathematical framework? i didn't find much material in the internet on problems related to rational triangles. In particular i'd like to know how is Gauss's solution to this problem related to number theoretical ideas (to which branch of number theory is it related?)?