Areas of primitive Heron Triangles that are perfect squares I am trying to find a sequence (in ascending order) of areas of primitive Heron Triangles that are perfect squares. For instance 36, 900, 7056, 44100 etc. These have been found by searching through generated integer triangles. But I am not sure if there exists other perfect squares in between these values that are also areas of primitive Heron triangles. Because of the sparseness of these numbers a simple search seems very inefficient. Does anyone know of a method of generating these numbers?
 A: In the system of equations:  $$\left\{\begin{aligned}&x+y=a+b\\&xy=qab\end{aligned}\right.$$
Another solution can be written.  $$a=p^2+2qps+(q^2-1)s^2$$  $$b=2s(p-(q-1)s)$$  $$x=2qs(p+(q-1)s)$$  $$y=p^2+2ps-(q^2-1)s^2$$  
All three formulas derived me just describe all solutions of the system. I think the question can be considered closed.
Solutions can be written as:  $$x=qs^2-qps$$  $$y=qps-p^2$$  $$a=ps-p^2$$  $$b=qs^2-ps$$
Or this:  $$x=qp^2+qps$$  $$y=qps+(q-1)s^2$$  $$a=qp^2+(2q-1)ps+(q-1)s^2$$  $$b=ps$$ 
If you meant these triangles Gerona. The formula I have written in this thread. Problem Heron of Alexandria.
They are still stubbornly ignore. Seeking solutions by brute force.
A: Well, the formula itself Geronova (Heronian) triangle.
$$S_g=\sqrt{(a+b+c)(a+b-c)(a-b+c)(b+c-a)}$$
If: $p,s,k,t$  -integers asked us. Then the solutions are.
$$a=(pt+ks)(k^2+t^2)ps$$
$$b=(pt-ks)((k^2+t^2)ps+(p^2+s^2)kt)$$
$$c=(pt+ks)(p^2+s^2)kt$$
$$S_g=4pskt(p^2t^2-k^2s^2)((k^2+t^2)ps+(p^2+s^2)kt)$$
A: An exhaustive list of the first 46 non-duplicated square areas of primitive Heronian triangles can be found at OEIS A248108. The list was generated by Sascha Kurz searching through lists of primitive Heronian triangles generated by him up to a maximum side length of 6,000,000.
