Non-Congruent Tilings - 9

Let me add a bit more to this earlier post in the non-congruent tiling series: http://nandacumar.blogspot.com/2018/11/non-congruent-tilings-of-plane-bit-more.html

This was the main question raised there (I still have no ideas):
If one looks for tiling the plane with non-congruent triangles of equal area and equal diameter (diameter of a triangle is its longest side) what happens?

Even if one requires that the mutually non-congruent triangles that tile the plane ought to have equal diameter and perimeter, the existence of such a tiling is not clear. Of course, one has an uncountable infinity of triangles all with same diameter and perimeter but whether one can tile the plane without gaps using a mutually noncongruent selection from them is not obvious.

More generally, one can ask if the mutually non-congruent tiles have all to be chosen from a one parameter family with the parameter values chosen from a finite and continuous 1D range (maybe with size 0 tiles disallowed), then, no tiling of the plane can happen.

And as usual, we can ask about non-Euclidean equivalents of these questions.