Proof of Hippocrates' Result

Skibidi vibes in geometry as we dive into Hippocrates' theorem. The center of the circle is lowkey the point D, chilling at the hypotenuse midpoint of triangle ABO. We are talking about a sigma moment where the diameter AC is double that of circle AEB. The smaller circle is straight-up half the area of the larger one, making the quarter circle AFBOA equal to the semicircle AEBDA. Subtracting the crescent area AFBDA from AFBOA leaves us with triangle ABO, while the lune comes from the semicircle. Triangle and lune are goon-level equal in area, proving the theorem's dankness!