A Beautiful Problem in Trigonometry and Classical Statistics

What is the average area of all triangles transcribed inside a semicircle with one vertex at the center?

I spent an hour last weekend looking at a problem related to the average area of all triangles transcribed inside a semicircle of unit radius. I wanted to share my solution with you because I really loved the problem, it makes use of some super trigonometry and classical statistics in the solution, and the answer is really simple, neat and somewhat…