TITLE: Most likely shape of random triangle
QUESTION [2 upvotes]: Observing cutting patterns in my wooden plate I came to the question: What is the expected (most likely) shape* of a triangle formed by three randomly chosen, but exactly in three points intersecting, lines?
Does the answer change if we limit the whole plane to a unit circle? Related but different questions are here and here.
*)The shape of the triangle shall be given by the ordered tupel of the two largest angles.
An intuitive geometrically derived solution would be $90°$ for the largest angle (the average value of the intervall $(0°,180°)$) and $45°$ for averaging all possible line positions given two other lines intersecting at $90°$.

REPLY [1 votes]: Hint:
Random lines over the whole plane is an undefined concept. But as you are only interested in angles, we can answer for random directions.
So we should draw three angles uniformly and look for the distribution of the largest and second largest cyclic difference. The computation is a little technical. It is a double integral where you keep the largest absolute difference fixed (or the second largest), and all cases are equiprobable. The expression of the domain seems a little complicated.
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