Editing Arbelos

{{Short description|Plane region bounded by three semicircles}}
[[File:Arbelos.svg|right|thumb|320px|An arbelos (grey region)]]
[[File:Arbelos sculpture Netherlands 1.jpg|thumb|Arbelos sculpture in [[Kaatsheuvel]], Netherlands]]

In [[geometry]], an '''arbelos''' is a plane region bounded by three [[semicircle]]s 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 [[diameter]]s.<ref name=wolfram/>

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.<ref name=archBLprop4/> 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 {{mvar|a}} and {{mvar|b}}; the third semicircle is [[Convex curve|convex]], with diameter {{mvar|''a''+''b''.}}<ref name=wolfram/> Let the diameters of the smaller semicircles be {{mvar|{{overline|BA}}}} and {{mvar|{{overline|AC}}}}; then the diameter of the larger semircle is {{mvar|{{overline|BC}}}}.

[[File:Arbelos diagram with points marked.svg|right|thumb|320px|Some special points on the arbelos.]]
<!--In the following sections, the corners of the arbelos are labeled {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, such that the diameter of the outer semicircle is {{mvar|BC}}, assumed to have unit length; and the diameters of the inner semicircles are {{mvar|AB}} and {{mvar|AC}}, assumed to have lengths ''r'' and 1−''r'', respectively.  The letter {{mvar|H}} denotes the point where the outer semicircle intercepts the line that is [[perpendicular]] to the diameter {{mvar|BC}} through the point {{mvar|A}}.-->

===Area===
Let {{mvar|H}} be the intersection of the larger semicircle with the line perpendicular to {{mvar|BC}} at {{mvar|A}}. Then the [[area (geometry)|area]] of the arbelos is equal to the area of a circle with diameter {{mvar|{{overline|HA}}}}.

'''Proof''': For the proof, reflect the arbelos over the line through the points {{mvar|B}} and {{mvar|C}}, and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles (with diameters {{mvar|{{overline|BA}}}}, {{mvar|{{overline|AC}}}}) are subtracted from the area of the large circle (with diameter {{mvar|{{overline|BC}}}}). Since the area of a circle is proportional to the square of the diameter ([[Euclid]]'s [[Euclid's Elements|Elements]], Book XII, Proposition 2; we do not need to know that the [[proportionality (mathematics)|constant of proportionality]] is {{math|{{sfrac|{{pi}}|4}}}}), the problem reduces to showing that <math>2|AH|^2 = |BC|^2 - |AC|^2 - |BA|^2</math>. The length {{mvar|{{abs|BC}}}} equals the sum of the lengths {{mvar|{{abs|BA}}}} and {{mvar|{{abs|AC}}}}, so this equation simplifies algebraically to the statement that <math>|AH|^2 = |BA||AC|</math>. Thus the claim is that the length of the segment {{mvar|{{overline|AH}}}} is the [[geometric mean]] of the lengths of the segments {{mvar|{{overline|BA}}}} and {{mvar|{{overline|AC}}}}. Now (see Figure) the triangle {{mvar|BHC}}, being inscribed in the semicircle, has a right angle at the point {{mvar|H}} (Euclid, Book III, Proposition 31), and consequently {{mvar|{{abs|HA}}}} is indeed a "mean proportional" between {{mvar|{{abs|BA}}}} and {{mvar|{{abs|AC}}}} (Euclid, Book VI, Proposition 8, Porism). This proof approximates the ancient Greek argument; [[Harold P. Boas]] cites a paper of [[Roger B. Nelsen]]<ref name="RBNelsen_2002">{{cite journal |last1=Nelsen |first1=R B |title=Proof without words: The area of an arbelos |journal=Math. Mag. |date=2002 |volume=75 |issue=2 |page=144|doi= 10.2307/3219152|jstor=3219152 }}</ref> who implemented the idea as the following [[proof without words]].<ref>{{cite journal| last=Boas | first=Harold P.| author-link1=Harold P. Boas | title=Reflections on the Arbelos | journal= [[The American Mathematical Monthly]]| year=2006| volume=113| issue=3| url=http://www.maa.org/programs/maa-awards/writing-awards/reflections-on-the-arbelos| pages=236–249 | doi=10.2307/27641891| jstor=27641891}}</ref>
[[File:Arbelos proof2.svg|center]]

===Rectangle===
Let {{mvar|D}} and {{mvar|E}} be the points where the segments {{mvar|{{overline|BH}}}} and {{mvar|{{overline|CH}}}} intersect the semicircles {{mvar|AB}} and {{mvar|AC}}, respectively.  The [[quadrilateral]] {{mvar|ADHE}} is actually a [[rectangle]].
:''Proof'': {{mvar|∠BDA}}, {{mvar|∠BHC}}, and {{mvar|∠AEC}} are right angles because they are inscribed in semicircles (by [[Thales's theorem]]). The quadrilateral {{mvar|ADHE}} therefore has three right angles, so it is a rectangle. ''Q.E.D.''

===Tangents===
The line {{mvar|DE}} is tangent to semicircle {{mvar|BA}} at {{mvar|D}} and semicircle {{mvar|AC}} at {{mvar|E}}.
:''Proof'': Since {{mvar|ADHE}} is a rectangle, the diagonals {{mvar|AH}} and {{mvar|DE}} have equal length and bisect each other at their intersection {{mvar|O}}. Therefore, <math>|OD| = |OA| = |OE|</math>. Also, since {{mvar|{{overline|OA}}}} is perpendicular to the diameters {{mvar|{{overline|BA}}}} and {{mvar|{{overline|AC}}}}, {{mvar|{{overline|OA}}}} is tangent to both semicircles at the point {{mvar|A}}. Finally, because the two tangents to a circle from any given exterior point have equal length, it follows that the other tangents from {{mvar|O}} to semicircles {{mvar|BA}} and {{mvar|AC}} are {{mvar|{{overline|OD}}}} and {{mvar|{{overline|OE}}}} respectively.

===Archimedes' circles===
The altitude {{mvar|AH}} 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 circle|inscribed]] in each of these regions, known as the [[Archimedes' circles]] of the arbelos, have the same size.

== Variations and generalisations ==
[[File:F-belos.svg|thumb|right|upright=1.0|example of an ''f''-belos]]
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.<ref>Antonio M. Oller-Marcen: [http://forumgeom.fau.edu/FG2013volume13/FG201310.pdf "The f-belos"]. In: ''Forum Geometricorum'', Volume 13 (2013), pp. 103–111.</ref>

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

==Etymology==
[[File:Arbelos Shoemakers Knife.jpg|right|thumb|upright=1.0|The type of shoemaker's knife that gave its name to the figure]]
The name ''arbelos'' comes from [[Ancient Greek|Greek]] ἡ ἄρβηλος ''he árbēlos'' or ἄρβυλος ''árbylos'', meaning "shoemaker's knife", a knife used by [[shoemaking|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==
<references>
<ref name=wolfram>{{Mathworld| title=Arbelos |urlname=Arbelos}}</ref>
<ref name=archBLprop4>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.'' ([https://web.archive.org/web/20080509094946/http://www.cut-the-knot.org/proofs/arbelos.shtml "Arbelos - the Shoemaker's Knife"])</ref>
</references>

==Bibliography==
* {{cite book | author = Johnson, R. A. | year = 1960 | title = Advanced Euclidean Geometry: An elementary treatise on the geometry of the triangle and the circle | edition = reprint of 1929 edition by Houghton Mifflin | publisher = Dover Publications | location = New York  | isbn = 978-0-486-46237-0 | pages = 116&ndash;117}}
* {{Cite book | authorlink = C. Stanley Ogilvy | last = Ogilvy | first = C. S. | year = 1990 | title = Excursions in Geometry | publisher = Dover | isbn = 0-486-26530-7 | pages = [https://archive.org/details/excursionsingeom0000ogil/page/51 51&ndash;54] | url = https://archive.org/details/excursionsingeom0000ogil/page/51 }}
* {{cite journal |last=Sondow | first = J. |arxiv=1210.2279 |title=The parbelos, a parabolic analog of the arbelos| journal = Amer. Math. Monthly |year=2013| volume = 120 | issue = 10 | pages = 929–935 | doi = 10.4169/amer.math.monthly.120.10.929 | s2cid = 33402874 }} [[American Mathematical Monthly]], 120 (2013), 929–935.
* {{cite book | last = Wells | first = D. | year = 1991 | title = The Penguin Dictionary of Curious and Interesting Geometry | url = https://archive.org/details/penguindictionar0000well/page/5 | url-access = registration | publisher = Penguin Books | location = New York | isbn = 0-14-011813-6 | pages = [https://archive.org/details/penguindictionar0000well/page/5 5&ndash;6] }}

==External links==
* {{commons category-inline}}
* {{wiktionary-inline}}

[[Category:Arbelos| ]]
[[Category:Greek mathematics| ]]
[[Category:Archimedes| ]]