Arbelos

The Arbelos (Greek Arbylos Άρβυλος for Schuster diameter) or the Sickle of Archimedes is a special, three semicircles limited geometrical figure. The famous Greek mathematician Archimedes is said to have studied the properties of the Arbelos and described in his Book of Lemmas.

The additionally drawn in the sketch circle coincides with the area Arbelos.

Evidence

Draw the auxiliary triangle. Side is the hypotenuse of the triangle, composed of the sections and. After the second set of Euclid, the square is above the height of the triangle is equal to the product of the two hypotenuse sections:

The circle whose diameter goes through and have the radius. The height of the triangle is so. Is the distance of the diameter of the large semi-circle. Is called the radius of the smaller semicircle and those of the smallest semi-circle, then. The radius of the large semicircle is therefore the half, ie.

After the set amount of Euclid applies: so.

With algebraic methods (ie, abstract work out - these were the Greeks not yet available ) you can quickly see that the assertion is true ( but you will win any insights into why this is so ). The area of ​​the Arbelos is equal to the area of the large semi-circle minus the area of the two small semicircles:

The area of ​​the circle which passes through and is. As shown above, applies after the second sentence of Euclid. It can thus be used in the formula for the area of ​​the Arbelos place now, this results in:

This proves that the area of ​​the Arbelos is equal to that of the circle which passes through and.