How common is an isosceles triangle?

 Isosceles triangle $t$ is one which has two sides with equal lengths, But it should not be equilateral. (Logically speaking every equilateral triangle is also isosceles in three different ways, but let us not be that logical). And talking about equality, we need a relative tolerance $\epsilon_{12}$ for any two sides and another relative tolerance $\epsilon_3$ for any third side. And it will be hard to dictate the mostly psychological limit when a triangle stops being equilateral and starts being an isosceles. 

Anyways, we are going towards a predicate $P(t,\epsilon_{12},\epsilon_3)$, which is true when the triangle is isosceles with given two limits.\[P(t,\epsilon_{12},\epsilon_3) \equiv |l_1 - l_2| < \epsilon_{12} l_{12} \text{ and } \epsilon_3 l_{12} < |l_{12} - l_3|\], where $l_{12}= (l_1+l_2)/2$ is the mean of any two edge lengths and $\epsilon_{12} \ll \epsilon_3$ and $\{l_1,l_2,l_3\}$ are the edge lenghts of the triangle $t$.  In plain English, two sides of $t$ are of an approx. equal length with a difference of $\epsilon_{12}$ at most, and the third edge length is a significantly different from those two. So, how often one encounters such a triangle? 

Img 1. An approximately isosceles triangle with shape tolerances $\epsilon_{12}$ and $\epsilon_3$.
The tolerance condition visualized as a red segment. 

To fix the psychological ratio value $\epsilon_3/\epsilon_{12}$ I conducted an experiment with a set of researchers (5 persons) in a distinguished University. It seems that $\epsilon_3\approx 5\epsilon_{12}$ is a rather good choice. Have to note that equilateral triangles are of second order rarity (prob(isosceles):prob(equilater) $\approx \mathcal{O}(\epsilon_{12}):\mathcal{O}(\epsilon_{12}^2)$.

Another formulation would define a predicate $P_3(.)$ for an equilateral triangle $t$:\[P_3(t,\epsilon_{12},\epsilon_3)\equiv \max_i(l_i) - \min_i(l_i)  < \epsilon_3 \text{mean}_i \, l_i\].Then, an isosceles would have a predicate $P_{12}(.)$:\[P_{12}(t,\epsilon_{12}) \equiv |l_1 - l_2| < \epsilon_{12} l_{12} \text{ and not } P_3(t,\epsilon_3)\] .  

One almost misses probabilistic geometry here... (Just as there is geometric algebra and algebraic geometry, there is also probabilistic geometry and geometric probability. The latter is -uhm- somewhat constructive and experimental, and the former theoretically solid, but somewhat obscure). So, let's go along the geometric probability road and make a choice of the topology of the probabilistic measure. Delaunay triangulation it is, and with it, the evident boundary effect. And the boundary effect depends on the number of points, when the PC shape is fixed (in this case, fixed to a unit square with a boundary of length 4). We have already outlined how to measure the 'boundariousness' of a PC, and how to define a boundary point. But now we do some census over the inhabitants of the Delaunay triangulation, approximating an infinite 2D plane with random points, at certain densities. 

But about the results: Below left is the ratio $R$ of observed isosceles from a set $P$ of points in a unit square. The small size of $P$ is revealed by fluctuations in different computations (top right corner). An indication that the $R$ really is of first order of the shape accuracy limit $\epsilon_{12}$, is at the right detail, where the ratio $R/\epsilon_{12}$ is shown to stay within a limited range while $\epsilon_{12}$ has a thousandfold range. 

Img. 2. Ratio $R$ of isosceles triangles in a Delaunay triangulation (left).  Visual proof of  $R\in\mathcal{O}(\epsilon_{12})$ behavior (right). The range of values stays within 1...45 while $\epsilon_{12}$ range is 1...1000.

A simplistic explanation for the category of $R$ is that the isosceles top vertex inhabits a narrow vertical band, when the best candidate for the base edge is scaled to unit length, see a sketch below. The band increases its width asymptotically towards infinity, and has some slight complexities at the spots where equilater triangles are. And long needle like triangles (those with the unit scaled presentation having the top vertex close to infinity) are more rare than others, but this does not get analyzed this time. The previous image is a better verification of the $T\in\mathcal{O}(\epsilon)$ behavior. 

Img. 3. Isosceles triangles (striped zone) with the base edge as a unit segment. The part below is symmetrical and omitted. A missing spot is the domain of equilateral triangles. 

There is something to scrutinize in Fig. 2. The Delauny process (should be familiar for the reader at this point) favors close to equilateral triangles because of the so called Delaunay property (every triangle $t$ defines an enclosing circle which has no other points within it than those of $t$), and maybe produces more isoclines, too? This is true, but requires a bit more finesse, and is left out of this text. And extension to 3D is obvious, but rather fruitless.