Arbelos

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 (connected), all on the same side of a straight line (the baseline) that contains their diameters.

[1] The earliest known reference to this figure is in Archimedes's Book of Lemmas, where some of its mathematical properties are stated as Propositions 4 through 8.

Let H be the intersection of the larger semicircle with the line perpendicular to BC at A.

Since the area of a circle is proportional to the square of the diameter (Euclid's Elements, Book XII, Proposition 2; we do not need to know that the constant of proportionality is ⁠π/4⁠), the problem reduces to showing that

Now (see Figure) the triangle BHC, being inscribed in the semicircle, has a right angle at the point H (Euclid, Book III, Proposition 31), and consequently |HA| is indeed a "mean proportional" between |BA| and |AC| (Euclid, Book VI, Proposition 8, Porism).

This proof approximates the ancient Greek argument; Harold P. Boas cites a paper of Roger B. Nelsen[3] who implemented the idea as the following proof without words.

[4] Let D and E be the points where the segments BH and CH intersect the semicircles AB and AC, respectively.

The parbelos is a figure similar to the arbelos, that uses parabola segments instead of half circles.

A generalisation comprising both arbelos and parbelos is the f-belos, which uses a certain type of similar differentiable functions.

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.