Thanks very much for the thoughts!
A variety of ideas is already presented here, so I don't have much to add besides this little experiment when I simulated 1,000,000 events (circle was simplified to 1,296,000 points) and evaluated the largest angle of every triangle:
The largest angle is a decisive factor for knowing the type of a triangle.
I found it interesting, because this interval of 90-180 degrees (meaning an obtuse triangle) is three times as big as interval of 60-90 degrees (meaning an acute triangle) while the ratio of P(obtuse) and P(acute) is also exactly three. So I wanted to see how the likelihood of those angles is distributed.