(a) Let ABC be an equilateral triangle, centered at O. A point P is chosen at random inside the triangle. Find the probability that P is closer to O than to any of the vertices. (In other words, find the probability that OP is shorter than AP, BP, and CP.)
(b) Let O be the center of square ABCD. A point P is chosen at random inside the square. Find the probability that the area of triangle PAB is less than half the area of the square.