Proof (Alhazen)
As we proved in the Hippocrates Proof, our goal is to find the area of the combined lunes and compare it to the area of the triangle. What shapes do you notice?
With those shapes engraved in our minds, what is the area of the (right) triangle?
We want to find the area of the red lune + green lune which will be the same area as the yellow triangle. We can do this by taking everything in the graphic and subtracting it by the big semi circle the triangle is inscribed in. ((Semi(red) + Semi(Green) + Tri(ABC)) - Semi(ACBA))=Area of Red Lune + Area of Green Lune.
What is the area of Semi(red)?
What is the area of Semi(green)?
What is the area of Semi(ACBA)?
Now put everything together! What do you notice?