Competition problem — Combinatorics

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Combinatorics Difficulty 4.8 AIME Prove it United States

Problem:

Four points, $A$ , $B$ , $C$ , and $D$ , are chosen randomly on the circumference of a circle with independent uniform probability. What is the expected number of sides of triangle $A B C$ for which the projection of $D$ onto the line containing the side lies between the two vertices?

Solution

Solution:

By linearity of expectations, the answer is exactly 3 times the probability that the orthogonal projection of $D$ onto $A B$ lies interior to the segment. This happens exactly when either $\angle D A B$ or $\angle D B A$ is obtuse, which is equivalent to saying that $A$ and $B$ lie on the same side of the diameter through $D$ . This happens with probability $1 / 2$ . Therefore, desired answer is $3 / 2$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.