Triangles inside circles

[bonanova]

A few puzzles posted in this forum have related to random triangles inside a circle.

By evaluating nasty integrals, or by my preferred method, simulation, it can be shown, perhaps surprisingly, that triangles constructed from sets of three uniformly chosen points within a circle cover only about 7.388% of the circle's area on average. After looking at

on the subject, I simulated 1 million triangles to determine the median area. It turns out to be about 5.335% of the circle's area. Read: a random triangle has a 50% chance of being smaller.

If the distribution of random-triangle areas has a mean of about 7.4% and a median of about 5.3%, what value might you expect for the mode?

[bonanova]

Amazingly, perhaps, the mode is zero.

Philosophically interesting: when sorted by size, more random triangles have zero area than any other size.

But that demands the question: how many triangles have zero area?

Why none, of course.

The probability of randomly choosing three coincident or collinear points is zero!

Pragmatically speaking, if we partition the possible area sizes into intervals, no matter how small,

and for this, see the previous post, the lowest-value interval will always characterize the greatest

number of random triangles.

[bonanova]

For mode:

Divide the area range from zero to the maximum possible area (inscribed equilateral triangle) into small intervals.

Which interval would contain the most area values for a large number of randomly drawn triangles?

[BMAD]

around 5.1% ??

[bonanova]

It's lower than that.

[bonanova]

Distribution of areas of 50,000 random triangles inside a circle.