What is a triangle? (Or what does the set of triangles "look like")

I recently had a discussion with friends on how much more common obtuse triangles are than acute. It depends of course on how you define a probability measure on triangles, and one has to make some assumptions. For some reasonable choices the answer is obtuse triangle are three times more common than acute. See for example the paper by Guy [1] and the discussion on Stack Exchange[2]. A figure from [1] is revealing. Suppose you break a stick in to three parts. They may or may not form a triangle depending it they satisfy the triangle inequality eg if a> b+c, but when they do they line in the middle triangle of the figure below. The curves are triples a,b,c of side lengths that satisfy Pythagoras' theorem (three hyperbolas) so give right angle triangles. The outside is obtuse and the inside is acute.

We see in this figure that a point represents a triple a,b,c, with a normalization a+b+c=1, and so we have a way to represent a triangle up to translation and similarity. So my question is what is a good way to think of "a triangle" modulo this equivalence relation. In a sense "what is a triangle". Stewart [3] gives a good answer to this. The "moduli space of triangles", and this turns out to be itself a triangle. In a way the "set of triangles is a triangle".

So in this case we consider the shaded area Phi as being the "representative" set of triangles, as others can be obtained by permutation of coordinates. Again the curve corresponds to right angle triangles. Now it is clear from the picture that in a uniform probability measure in these coordinates the obtuse triangles are more common.  

References
[1] Guy, Richard K. "There are three times as many obtuse-angled triangles as there are acute-angled ones." Mathematics Magazine 66.3 (1993): 175-179.
[2] brick (https://math.stackexchange.com/users/187522/brick), Probability of triangle to be acute?, (version: 2015-04-24)
[3] Stewart, Ian. "Why do all triangles form a triangle?." The American Mathematical Monthly 124.1 (2017): 70-73