(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.

Stanry Jun 29, 2024