We add a little to Tiling the plane with pairwise non-congruent rational triangles
Question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same area? If "yes", can such an equal area tiling be achieved with the perimeters of the triangles upper bounded?
Remark 1: For any given possible area A, https://nyjm.albany.edu/j/1998/4-1p.pdf proves that there are indeed infinitely many rational triangles all with area A. But from among triangles for a given A to select a set that tiles the plane might not be possible. If the answer to the question is "no", one could ask if mutually non-congruent rational triangles whose areas are all one of a finite set of values would tile the plane.
- I couldn't figure out if the theorem(s) proved in https://web.archive.org/web/20210414080312/https://math.dartmouth.edu/~carlp/PDF/paper14.pdf are relevant here.
Further question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same perimeter?
