1. L UNES OF HIPPOCRATES History 310. MW: 11:00am Fall 2014 Professor: Dr. Robert Mena by Noor Shukairy Israel Flores ThuyNguyen Nguyen

2. L UNES OF H IPPOCRATES Lunes of Hippocrates states the area of the two lunes outside of the semicircle is the same as the area of the right triangle inscribed inside of that semicircle.

3. Step 1: Start with a Semi-circle

4. Step 2 : Inscribe a triangle inside of the semi-circle. By Thales’ theorem, any triangle inscribed in a semicircle that shares the hypotenuse with the diameter is a right triangle.

5. Step 3: Inscribe a semi-circle on each leg of the triangle.

6. Now that we have the Lunes desired, we can begin to prove it geometrically.

7. Note : Geometrically, the big semi-circle is equal to the triangle and two slivers.

8. Also, the two half circles on the legs of the triangle can be broken down into the Lunes and slivers depicted in previous slides.

11. A LGEBRA PROOF Given a semicircle with a right triangle inscribed on its diameter and a semicircle on each of the sides of the inscribed right triangle, we will show that the area of the two lunes outside of the large semicircle is the same as the area of the inscribed right triangle.