MHB What is the Probability of Distances in an Equilateral Triangle?

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The discussion focuses on determining the probability that a randomly chosen point P within an equilateral triangle T has a corresponding point Q within T that is more than the altitude of T away from P. The altitude of an equilateral triangle can be calculated using its side length. The suggested solution involves geometric probability and the properties of distances within the triangle. Participants explore the implications of the triangle's symmetry and uniform distribution in calculating the desired probability. The conversation emphasizes the mathematical principles involved in solving this geometric probability problem.
lfdahl
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A point $P$ is chosen at random with respect to the uniform distribution in an
equilateral triangle $T$. What is the probability that there is a point $Q$ in $T$ whose distance
from $P$ is larger than the altitude of $T$?
 
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