What is the modern context of Gauss's work on triangles with integer sides and circumradius?

Question

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?)?

Answer 1

"Curtius's problem" is Schumacher's personal name repeated only by Dickson. By scaling, one can reduce the problem to that about triangles with rational sides and the circumradius, which is equivalent to the more commonly considered problem about triangles whose sides and the area are rationals, since $r=abc/4A$. These, or the version where the sides and the area are integers, are now called Heron triangles. The solution, for the integer version, goes back to Brahmagupta, and one can see that Gauss's formulas are related to Brahmagupta's (who stated them in words, of course): $$a=n(m^2+k^2);\ \ b=m(n^2+k^2);\ \ c=(m+n)(mn-k^2).$$ These formulas, just like those for Pythagorean triples, relate Heron triangles to Diophantine equations. The long known connection between Diophantine equations and elliptic curves was made particularly prominent in the Wiles's proof of the Last Fermat theorem, and elliptic curves provide the most common perspective on Heron triangles in recent research, see e.g. Heron Triangles and their Elliptic Curves by Halbeisen-Hungerbuhler. They are also quite popular in recreational mathematics, and e.g. Yiu's survey relates them to Gaussian integers. But I did not see a connection explicitly made to Curtius's version of the problem, or Gauss's contribution to it.

More remotely, with the additional condition that the triangle be right angled, this leads to the still unsolved problem of congruent numbers. The problem goes back to al-Khazin (c. 940 AD), and a number is congruent when it is the area of such a triangle. A simple criterion for a number to be congruent would follow from a very famous conjecture, also about elliptic curves, the Birch and Swinnerton-Dyer conjecture. It is one of the millennium problems.