Ponder This 十月份
我差点已经彻底的忘掉这个东西了.
Consider 3 points chosen at random in the interior of a triangle. We ask what is the expected (average) area of the triangle determined by these 3 points as a fraction of the area of the triangle they are chosen in. It is fairly easy to see that this fraction is independent of the size or shape of the original triangle. So we may assume we are choosing points in the triangle, S, in the x-y plane with vertices (0,0), (1,0) and (0,1). The value we are asking for can be computed by answering the following questions.
1. Consider the minimum triangle (with edges parallel to the edges of S), T, containing the 3 random points. What is the ratio of the expected area of T to the area of S.
2. There are 2 configurations of T and the 3 random points which occur with positive probability. These are:
- Case A - One random point is a vertex of T, another lies along the opposite side and the third is in the interior of T.
- Case B - Each edge of T contains one of the 3 random points.
What is the probability of occurrence of each of these cases?
3. What is the expected fraction of the area of T which is inside the triangle for each of the cases A and B above?
4. Combining the answers to questions 1-3, what is the ratio of the expected area of a triangle, T, formed by choosing 3 points at random in another triangle, S, to the area of S.
