The Lune of Hippocrates Puzzle

This ancient puzzle from Greek mathematics asks for the area of the cresecent shaped figure compared to that of the triangle. We'll solve the puzzle using compass and straight-edge (synthetic) geometry. Michael Penn's solution (a different approach): https://youtu.be/w4MGPq_ilDU ... https://www.youtube.com/watch?v=-Ii0Dtfw67E