Basic question: Can the Euclidean plane be divided into a vertex-to-vertex arrangement of non-overlapping triangles such that every edge has a unique rational length that lies between 1 and some specific rational R greater than 1?
(By edge-to-edge layout, we mean any vertex of the layout is necessarily an end point of all the edges meeting at that vertex. IOW, exactly two triangles meet at each edge in the layout. Also note that the length of each edge has to be different from other edge lengths.)
If this is possible, one can apply further constraints such as "all triangles should have equal area (OR equal perimeter)". Alternatively, one can relax the vertex-to-vertex requirement. One can also replace the requirement that every edge has a unique length with (say) the triangles being pairwise non-congruent.
Note: Requiring the lengths of all edges to be integers rather than rationals would lead to the lengths of the edges being unbounded even if a triangulation with all edges having unique lengths is possible (not sure if this is possible).