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?
(In reply to Not faster, simpler, or better
by Steve Herman)
I agree that I have no proof, nor for that matter even a clear view of the problem. There are presumably an infinitude of points within any closed figure. If the problem specified integer dimensions of the square, and required integer coordinates of the three points, there could be a specific problem. Must the three points be inside the square, or may they be on the square? If the center is on the triangle, is that considered "within"? What exactly does "random" mean in this case? Presumably (a) any point is equally likely, and (b) the second and third are obtained independently of the first and second. No person placing dots (as problem specifies) would be placing them randomly, especially if he knew the question (might be the "one-corner" painter). I hope the rest of you can convince each other that you are solving the same problem; I'll sit this one out.