Are there any interesting results on optimal packings in the plane using a fixed number of triangles (without a fixed size or shape constraint)?

For instance, what's the maximum area packing of the unit circle using at most $N$ non-intersecting (except possibly on the boundary) triangles inside the circle? This isn't very difficult with the additional assumption that the vertices of the triangle all lie on the circumference (we can smooth the triangles "together" to form a triangulation of a $(N+2)$-gon, and either use Jensen or compactness and further smoothing to show the regular $(N+2)$-gon is optimal). But in the general case, smoothing seems much more difficult---the triangles can "block" each other in much more complicated configurations, which seem hard to classify.

**Edit (8/18/14).** First, a potentially fruitful (and quite nice!) idea from fedja copied from AoPS, which works for the **semicircle** case (i.e. packing into a semicircle rather than a circle):

Then [in the semicircle case] all you would need to do (assuming that the diameter is $[-1,1]$) would be to consider the vertical cross-sections and notice that their length is a piecewise linear function with at most as many "down" corners as we have triangles, dominated by $\sqrt{1-x^2}$, so we would have to estimate the area between the axis and a piecewise convex curve with a fixed number of pieces under the circular arc, and everything would become pretty obvious. I guess the idea has a good chance to work in the current setting too, but the details will, most likely, be much less elegant :(.

I might as well mention the corresponding MSE thread.

Also, here's a potentially serious obstruction to most smoothing arguments, from an email of Tim Chu's:

Take a small $N$-gon and extend the exterior angles.

Place it in at the center of a very big circle. Then you get a shape that's almost the same as if you made a regular $N$-gon by having all the triangles share one vertex at the center of the circle, except instead these $N$ triangles trace out a small $N$-gon in the center instead of meeting at a single point.

The problem is that in this arrangement, you can't smooth any single triangle to have a greater area [which is what the easiest smoothing approaches try to do]. So in a sense, this arrangement is "locally optimal" but not "globally optimal". In this case, it's rather clear that the example presented isn't actually optimal (namely, its area is bounded above by that of the $N$-gon), but there are other examples in this vein ("locally optimal" but not "globally optimal") in which it's not clear why they don't beat the $(N+2)$-gon.

(This example also breaks any approach that attempts to directly find two triangles $ABC$ and $DEF$ such that no other triangles intersect the convex hull of $ABCDEF$.)