Heron scalene triangles can be constructed from either combining two Pythagorean triangles or by subtracting two Pythagorean triangles.
I provided a method in Generating scalene Heron triangles — Part 1
Now, let’s construct Heron scalene triangles by subtracting two Pythagorean triangles.
AB = c
Here are all scalene Heron triangles when the shortest leg is
There are also the indecomposable herons that have rational heights for example the {25, 34, 39, 420} triangle {a, b, c, area}. So I do think your statement that all triangles can be constructed above is a bit misleading.
P.
Right. For the indecomposable ones, we need to present an alternative way by using half-angle formulae
I’ll post some ideas on a future blog