Suppose you are given a "random" triangle. It does have a longest side. Call it n, and its enpoints a and b. Call the third point of the triangle c. c / \ / \ /________ a n b
What is wrong with the assumption that the distances from a to c and the distances from b to c are independent, and that the distance from a to c (and likewise the distance from b to c) is uniformly distributed over (0,n]? (ie the ratios of the two other sides to the longest side should be uniformly distributed over (0,1]) These seem like viable, intuitive assumptions about a random triangle. If not, then in what way should the lengths of the sides be dependent?
The reason I ask is because these simple assumptions alone (not assumptions involving random points in R^2 and the like) allow the construction of a measure space of triangles that does not give 3/4 as the probability of obtuseness. If 3/4 is the accepted answer, then this implies that a "random" triangle follows nonuniform distributions in the ratios of its sides. This doesn't make sense to me.