A rational triangle is one in which all side lengths are rational numbers.

**Question:** Can we tile the Euclidean plane with rational triangles that are pairwise non-congruent? No further requirements on the triangles. Their perimeters are not upper or lower bounded.

The answer seems "no" but I have no proof. It is known that there is no tiling of the plane by pairwise non-congruent triangles all of same area and same perimeter (https://arxiv.org/abs/1711.04504) - not sure if that is relevant here.

Further, we can ask if the plane can be tiled by quadrangles and so forth whose sides are all rational (and if the answer is "yes", add further constraints on diagonals etc.).

mightlead to an answer to the present question - with the added benefit that every edge on every triangle has a unique length. The downside is that the triangles would be unbounded in size! $\endgroup$