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Three Points in a Square (Posted on 2008-03-17) Difficulty: 3 of 5
Three points are chosen at random inside a square. Each point is chosen by choosing a random x-coordinate and a random y-coordinate.

A triangle is drawn with the three random points as the vertices. What is the probability that the center of the square is inside the triangle?

No Solution Yet Submitted by Brian Smith    
Rating: 3.2000 (5 votes)

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Some Thoughts Not faster, simpler, or better | Comment 9 of 18 |
The average area of a random triangle in the unit square is 11/144, significantly less than 1/4.   This value comes from Mathworld (see my earlier reference).  It is not surprising that the probability of a random triangle containing the center of the square is a lot bigger then the average area of a random triangle.  For instance, the maximum area of a triangle inside a square is 1/2 (just as Ed suggested).    Consider the set of all embedded triangles whose area is 1/2.  Their average area of 1/2 is a lot less than the probability that they contain the center, which turns out to be 100%.

Edited on March 18, 2008, 12:55 am
  Posted by Steve Herman on 2008-03-18 00:54:09

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