I think you're on the right track with this one. I have some reasoning now using that:

@kdkeller @xpil Say we are looking for a triangle T area A and rational sides x,y,z

Let there be a Pythagorean Triple a,b,c and their triangle have area Δ

T exists if such a triple exists that

A=Δs²

x=as, y=bs, z=cs

Where s (scale) must contain the LCM of a,b,c as a factor.

This is still quite new to me, so I can't be too sure about my reasoning. I don't know either if cases outside this can or can't exist.