perplexus.info :: Probability : 4-gon Probability

4-gon Probability (Posted on 2006-11-03)

Points A, B, C, and D (no three of which are collinear) lie in a plane. If point D lies randomly within ΔABC's anticomplementary triangle, then what is the probability that 4-gon ABCD is (concave, convex, reflex)?

What would be the answer if ΔABC is equilateral and "ΔABC's anticomplementary triangle" is replaced with "ΔABC's circumcircle"?

NOTE: The anticomplementary triangle of a given triangle is formed by three lines. Each line passes through a vertex of the given triangle and is parallel to the opposite side.

NOTE: The circumcircle of a given triangle is the unique circle which passes through each of its vertices.

Submitted by Bractals

Rating: 3.2500 (4 votes)

Solution: (Hide)

If A'B'C' is the anticomplementary triangle of ΔABC; then 4-gon ABCD is concave if point D lies in the interior of ΔABC, convex if point D lies in the interior of ΔAB'C, and reflex if point D lies in the interior of ΔABC' or ΔA'BC. Therefore, the probabilities are (¼,¼,½).

The 4-gon ABCD is concave if the point D lies in the interior of ΔABC, convex if point D lies in the interior of one of the circular segments, and reflex if point D lies in the interior of one of the other two circular segments. Therefore,
3r²√3 ------- 4 Concave: --------- πr² 1 - probability(concave) Convex: -------------------------- 3 Reflex: 2 probability(convex)
Thus, approximately (0.1955,0.4135,0.3910)

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Part 1 is simpler.	Jer	2006-11-03 11:37:43
outline of part 1; solution part 2	Charlie	2006-11-03 09:55:11

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