Let $a,b,c$ be integers which are the sides of a triangle with integral area, a so called Heronian triangle. This website attributes to Gauss the result that there must then exist integers $m,n,p,q$ such that

$a = mn(p^2+q^2)$

$b = (mp)^2+(nq)^2$

$c = (m+n)(mp^2-nq^2)$

(where I left out a $4pq$ factor designed to make the radius of the circumscribed circle integral as well). It's not hard to see that the triangle defined by these formulas is indeed Heronian, however I could neither prove nor find a reference for the fact that this parametrization is exhaustive.

Can someone do one of these two things?

Thanks!

(Note: I'm communicating this question on behalf of my dad, who is really the person who looked into that but is not easily capable to asking it himself. I may be slow to respond on his behalf if questions come up).

Let $a,b,c$ be integers which are the sides of a triangle with integral area, a so called Heronian triangle. This website attributes to Gauss the result that there must then exist integers $m,n,p,q$ such that

$a = mn(p^2+q^2)$

$b = (mp)^2+(nq)^2$

$c = (m+n)(mp^2-nq^2)$

(where I left out a $4pq$ factor designed to make the radius of the circumscribed circle integral as well). It's not hard to see that the triangle defined by these formulas is indeed Heronian, however I could neither prove nor find a reference for the fact that this parametrization is exhaustive.

Can someone do one of these two things?

Thanks!

(Note: I'm communicating this question on behalf of my dad, who is really the person who looked into that but is not easily capable of asking it himself over here. I may be slow to respond on his behalf if questions come up).