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?
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