Let's say you have a square, say $ABCD$, with side length 1. A point $P$ is randomly chosen from inside the square, with any point being equally probable of being chosen. What is the probability that the angle $\angle APB$ is a right angle?
This looks simple, and I do this problem for the case of $\angle APB$ being acute or obtuse, and that the set of all points satisfying the condition above is a semicircle inside the square centered at the midpoint of $AB$. How do you do this, or is it impossible?