After all the discussion raised by this old question, I am wondering about a somewhat complementary one:

For any given rectangle, does there exist a finite set of pairwise

differentisosceles triangles which tile it?

It is easy to tile e.g. a $1\times a$ rectangle for $1<a<2$ by four isosceles triangles, but with two of them being equal. In the case that $a=\sqrt{\frac{5-\sqrt{5}}2}$, we are lucky and can split one of those into two smaller ones, obtaining a tiling into 5 different isosceles triangles (with all occurring angles being multiples of $\frac\pi{10}$). BTW, we can iterate that by splitting the blue triangle again etc., getting tilings of the same rectangle into $k$ different isosceles triangles for all $k\ge5$.

I am quite sure the answer to the initial question is no, and it may even be interesting to restrict it to the following:

For which other rectangles is such a tiling known to exist?

And possibly, it doesn't even make a difference if we allow an *infinite* set of pairwise different isosceles triangles!