Suppose we are given $n$ distinct points $x_1, \dots, x_n \in \mathbb{S}^1$ on the unit circle in $\mathbb{R}^2$. Any three points induce a triangle $\Delta(x_i, x_j, x_k)$ and a total of $\sim n^3/6$ triangle can be obtained this way. Naturally, some of these triangles can have the same area but it's also clear that not all of them will have the same area. I am interested in how few different areas there can be.
Question
Are there always $\gtrsim n^2$ different areas?
First Motivation Usually, for questions of this type, I would expect the extremal configuration to be pretty rigid. Here, one would naturally expect equispaced points to be optimal and equispaced points do indeed lead to $\sim n^2$ different areas which can be seen as follows: fix a point, then there are $2$ points left to choose. If every possible choice leads to a triangle with unique area, we get $\sim n^2/2$ different areas this way and by rotational symmetry, it didn't matter what the first point was.
Now something funny happens: if $\alpha > 0$ is an irrational number and we consider the irrational rotations on the torus $$ x_k = (\cos(2 \pi k \alpha), \sin(2 \pi k \alpha)),$$ then the first $n$ elements of this infinite sequence are another example of a set of points that only induces $\sim n^2$ different areas. The argument is simple: the (Euclidean) distance between any two of these points $$ \| x_k - x_m \| = |e^{2\pi i k \alpha} - e^{2 \pi i m \alpha}| = |e^{2 \pi i (k-m) \alpha} -1 | $$ is at most one of $n$ different values. For a triangle with all three points on $\mathbb{S}^1$, the area is uniquely determined from knowing two of the sidelengths (since that determines the third) and thus there are at $\lesssim n^2$ different areas. So this is somehow curious: there are several examples that, up to constants, only give $\sim n^2$ different areas. Of course all these examples are still pretty rigid.
Second Motivation Unless I messed up the computation, the area of the triangle spanned by $(\cos s, \sin s), (\cos t, sin t)$ and $(\cos u, \sin u)$ is $$ \mbox{area} [\Delta(x_i, x_j, x_k)] = 4 \sin\left( \frac{s-t}{2} \right)^2 \sin\left( \frac{u-s}{2} \right)^2 \sin\left( \frac{t-u}{2} \right)^2 $$ which is somehow reminiscent of basic additive combinatorics with a nonlinear twist. The examples above are also essentially arithmetic progressions, so...
Extensions The same argument should show that when looking at the convex hull of all $k-$element subsets (of which there are $\sim n^k/k!$), these constructions have $\lesssim n^{k-1}$ distinct areas and maybe that is also extremal.
