The obstacle points shown above are equally spaced around a circle of radius 0.32096.... Given these obstacle points, the greatest area I can find for a valid triangle is 0.82863..., attained by the two triangles shown below.

(Reader Matt found these as well.)

I call the first triangle "the edge triangle," and I call the second triangle "the spotlight triangle."

The reason I placed the obstacle points a distance 0.32096... from the center is that doing so gives the edge triangle and the spotlight triangle equal areas. If the obstacle points were to move closer to the center, the area of the edge triangle would increase and the area of the spotlight triangle would decrease; if the obstacle points were to move further from the center, the area of the spotlight triangle would increase and the area of the edge triangle would decrease. The greater of the two areas is minimized when the obstacle points are as shown. It is on the strength of that circumstance that I guessed these obstacles as minimizing the maximum area over valid triangles.

The reason I placed the obstacle points a distance 0.32096... from the center is that doing so gives the edge triangle and the spotlight triangle equal areas. If the obstacle points were to move closer to the center, the area of the edge triangle would increase and the area of the spotlight triangle would decrease; if the obstacle points were to move further from the center, the area of the spotlight triangle would increase and the area of the edge triangle would decrease. The greater of the two areas is minimized when the obstacle points are as shown. It is on the strength of that circumstance that I guessed these obstacles as minimizing the maximum area over valid triangles.

Anyhow, recently I had a bit of downtime, so I wrote a quick program to look randomly for valid triangles with area greater than 0.82863.... To do that, I first generated 1,000 points distributed randomly over the disk:

These thousand points determine 166,167,000 triangles. I went through those triangles, discarding any of them with area less than 0.815. What remained was a list of about 1.6 million large triangles. I checked this list to see if any of the large triangles were valid. However, none of them were.

So, this particular attempt at beating the record 0.82863... failed.

Here are some of the large triangles that were closest to being valid:

It looks as if these triangles are "trying" to be like the spotlight triangle, with one case of an edge triangle. (I think edge triangles are less well represented because they have fewer ways to avoid the obstacles without losing area rapidly by sacrificing altitude.)

Anyway, this exercise increases my confidence somewhat that the edge triangle and the spotlight triangle are indeed maximizers for the obstacle points shown, and that up to symmetry these two triangles are the only maximizers.

Of course, none of this is a proof of anything.

Anyway, this exercise increases my confidence somewhat that the edge triangle and the spotlight triangle are indeed maximizers for the obstacle points shown, and that up to symmetry these two triangles are the only maximizers.

Of course, none of this is a proof of anything.

Some notes on the program:

- Calculating area is much faster than checking validity. That's why I calculated 166,167,000 triangle areas first, and then checked only the largest triangles for validity. It would have been impractically slow to work in the reverse order (checking 166,167,000 triangles for validity and then calculating the areas for the subset found to be valid). Calculating 166,167,000 triangle areas took about 12 hours. (If I do this again, I'll see if I can optimize the area calculation.)
- In the validity test, I'm using barycentric coordinates to determine whether a given point is contained by a given triangle. I'm not sure the barycentric test is faster than one based on same-side-of-line considerations, but so far I haven't needed to worry about speed in this phase of things. (Also, barycentric coordinates make it easy to be quantitative in various ways, for example identifying triangles that are close to valid, as shown in the animation above.)
- 166,167,000 triangles is a lot of triangles, so the program never holds all of them in memory. Instead of generating all those triangles ahead of time, I put the area test inside a threefold-nested loop over the thousand randomly selected points. Triples of points passing the area test were added to a steadily growing list of large triangles.