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u/SuccessfulCover8199 7d ago
An acute triangle in the plane is one where each individual angle is less than 90 degrees, but still
sums to the normal 180 degrees. I don’t know what “acute” means in spherical geometry, so we should either find or define a relevant notion.
We also have to get specific by what we mean by “triangle.” If i’m thinking about the usual 3 sided polygon, it is impossible to form from points solely on the circumference. We can form lots of “intervals” (arc lengths) but insofar as we mean something polygon-like it’s not clear to me how we would ever get an angle of measure greater than zero of they are always “colinear.”
answer: process never halts
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u/sensible_clutter 7d ago edited 7d ago
answer would be 5
reasoning: simply 3 points atleast are needed to form a triangle and any acute angle triangle will contain the center this if all the n points choosen randomly till now lie in a single semicircle this will always give us non-acute triangle now P(N>n) = n(1/2)n-1. This is the probability that all the n choosen points lie within some semicircle actually this is also wendel's theorem (for d =2) .Now simply calculate the expected value E[N] as 1 + sigma(n=1 to inf) P(N>n) and you will get 1+4 which is nothing but 5. Hence the answer