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.

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. The word ''arbelos'' is Greek for 'shoemaker's knife'. The figure is closely related to the Pappus chain.

Properties

Two of the semicircles are necessarily concave, with arbitrary diameters and ; the third semicircle is convex, with diameter

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 (with diameters , ) are subtracted from the area of the large circle (with diameter ). 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 ), the problem reduces to showing that $2, AH, ^2 = , BC, ^2 - , AC, ^2 - , B, ^2$. The length equals the sum of the lengths and , so this equation simplifies algebraically to the statement that $, AH, ^2 = , BA, , AC,$. Thus the claim is that the length of the segment is the geometric mean of the lengths of the segments and . Now (see Figure) the triangle , being inscribed in the semicircle, has a right angle at the point (Euclid, Book III, Proposition 31), and consequently is indeed a "mean proportional" between and (Euclid, Book VI, Proposition 8, Porism). 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.
:''Proof'': , , and are right angles because they are inscribed in semicircles (by Thales's theorem). The quadrilateral therefore has three right angles, so it is a rectangle. ''Q.E.D.''

Tangents

The line is tangent to semicircle at and semicircle at .
:''Proof'': Since is a right angle, equals minus . However, also equals minus (since is a right angle). Therefore triangles and are similar. Therefore equals , where is the midpoint of and is the midpoint of . But is a straight line, so and are supplementary angles. Therefore the sum of and is π. is a right angle. The sum of the angles in any quadrilateral is 2π, so in quadrilateral , must be a right angle. But is a rectangle, so the midpoint of (the rectangle's diagonal) is also the midpoint of (the rectangle's other diagonal). As (defined as the midpoint of ) is the center of semicircle , and angle is a right angle, then is tangent to semicircle at . By analogous reasoning is tangent to semicircle at . ''Q.E.D.''

Archimedes' circles

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 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.Antonio M. Oller-Marcen
"The f-belos"
In: ''Forum Geometricorum'', Volume 13 (2013), pp. 103–111.

In the Poincaré half-plane model of the hyperbolic plane, an arbelos models an ideal triangle.

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.

See also

* Archimedes' quadruplets
* Bankoff circle
* Schoch circles
* Schoch line
* Woo circles
* Pappus chain
* Salinon

References

Thomas Little Heath (1897), ''The Works of Archimedes''. Cambridge University Press. Proposition 4 in the ''Book of Lemmas''. Quote: ''If AB be the diameter of a semicircle and N any point on AB, and if semicircles be described within the first semicircle and having AN, BN as diameters respectively, the figure included between the circumferences of the three semicircles is "what Archimedes called arbelos"; and its area is equal to the circle on PN as diameter, where PN is perpendicular to AB and meets the original semicircle in P.''
"Arbelos - the Shoemaker's Knife"

Bibliography

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* American Mathematical Monthly, 120 (2013), 929-935.
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External links

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