Obtuse Random Triangles from Three Parts of the Unit Interval

Two points are chosen at random in the unit interval. From the resulting three parts, we try to form a triangle. If a triangle can be formed, we are interested in whether it is obtuse or not. It is known that a triangle can be formed with probability 1/4. An obtuse triangle can be formed with probability . If a triangle can be formed, it is obtuse with probability .


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Snapshot 1: Only one-quarter of the experiments result in a triangle. This snapshot shows a typical example where a triangle cannot be formed: one of the three parts is longer than the sum of the lengths of the other two parts.
Snapshot 2: Here is another example where a triangle cannot be formed.
Snapshot 3: If a triangle can be formed, it is obtuse with quite a high probability (0.682).
Snapshot 4: Approximately one-third of the cases where a triangle can be formed are such that the triangle is not obtuse.
The probability that the random triangle is obtuse is calculated in [1]. The problem is also considered in [2, pp. 31–32].
In another Demonstration, we consider the related problem of generating random triangles in a rectangle; see the Related Links.
[1] M. G. C. Holterman, "Solution of Problem 858: Probability of an Obtuse Triangle," Mathematics Magazine, 46(5), 1973 pp. 294–295.
[2] P. J. Nahin, Digital Dice: Computational Solutions to Practical Probability Problems, Princeton, NJ: Princeton University Press, 2008.
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