Lune of Hippocrates

With the lune of Hippocrates, which are the Greek mathematician Hippocrates of Chios attributed ( around 450 BC ), one could prove in ancient Greece that also curvilinear surface elements can be calculated by rational numbers.

Evidence

According to the Pythagorean theorem, the sum of the areas of Kathetenquadrate of a right triangle is equal to the area of Hypotenusenquadrats. After the generalized Pythagorean theorem, this relationship also applies to other figures similar to each other. For semicircles this means that the cumulative area of ​​the semicircles on the other sides equal to the area of the semicircle on the hypotenuse (step 1).

Flips you the semicircle on the hypotenuse, so this overlaps with the two Kathetenhalbkreisen, the circle arc to the theorem of Thales through the point C is ( step 2).

By removing the overlapping circular segments (step 3), remain the Hypotenusenhalbkreis the triangle itself and the two Kathetenhalbkreisen the two crescent-shaped outer circle parts that Halfmoon.

The following applies:

And

From

Then follows:

Variants

There are many different variations and possibilities of generalization of the Pythagorean theorem and the lune of Hippocrates. Next to the right-angled triangle, the aforementioned following square, over the four sides of a square in each case a half-moon, a further example.