In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others, all on the same side of a straight line that contains their diameters. The earliest known reference to this figure is in the Book of Lemmas, where some of its mathematical properties are stated as Propositions 4 through 8. The word arbelos is Greek for 'shoemaker's knife'.
Properties
Two of the semicircles are necessarily concave, with arbitrary diameters a and b; the third semicircle is convex, with diametera+b.
Area
The area of the arbelos is equal to the area of a circle with diameter. Proof: For the proof, reflect the arbelos over the line through the points and, and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles are subtracted from the area of the large circle. Since the area of a circle is proportional to the square of the diameter, the problem reduces to showing that. The length equals the sum of the lengths and, so this equation simplifies algebraically to the statement that. Thus the claim is that the length of the segment is the geometric mean of the lengths of the segments and. Now the triangle, being inscribed in the semicircle, has a right angle at the point , and consequently is indeed a "mean proportional" between and . This proof approximates the ancient Greek argument; Harold P. Boas cites a paper of Roger B. Nelsen who implemented the idea as the following proof without words.
Rectangle
Let and be the points where the segments and intersect the semicircles and, respectively. The quadrilateral is actually a rectangle.
Tangents
The line is tangent to semicircle at and semicircle at.
The altitude divides the arbelos into two regions, each bounded by a semicircle, a straight line segment, and an arc of the outer semicircle. The circles inscribed in each of these regions, known as the Archimedes' circles of the arbelos, have the same size.
Variations and generalisations
The parbelos is figure similar to the arbelos, that uses parabola segments instead of half circles. A generalisation comprising both arbelos and parbelos is the f-belos. that uses a certain type of similar differentiable functions.
Etymology
The name arbelos comes from Greek ἡ ἄρβηλος he árbēlos or ἄρβυλος árbylos, meaning "shoemaker's knife", a knife used by cobblers from antiquity to the current day, whose blade is said to resemble the geometric figure.
