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.<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.
PropertiesEdit
Two of the semicircles are necessarily concave, with arbitrary diameters Template:Mvar and Template:Mvar; the third semicircle is convex, with diameter Template:Mvar<ref name=wolfram/> Let the diameters of the smaller semicircles be Template:Mvar and Template:Mvar; then the diameter of the larger semircle is Template:Mvar.
AreaEdit
Let Template:Mvar be the intersection of the larger semicircle with the line perpendicular to Template:Mvar at Template:Mvar. Then the area of the arbelos is equal to the area of a circle with diameter Template:Mvar.
Proof: For the proof, reflect the arbelos over the line through the points Template:Mvar and Template:Mvar, and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles (with diameters Template:Mvar, Template:Mvar) are subtracted from the area of the large circle (with diameter Template:Mvar). 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 Template:Math), the problem reduces to showing that <math>2|AH|^2 = |BC|^2 - |AC|^2 - |BA|^2</math>. The length Template:Mvar equals the sum of the lengths Template:Mvar and Template:Mvar, 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 Template:Mvar is the geometric mean of the lengths of the segments Template:Mvar and Template:Mvar. Now (see Figure) the triangle Template:Mvar, being inscribed in the semicircle, has a right angle at the point Template:Mvar (Euclid, Book III, Proposition 31), and consequently Template:Mvar is indeed a "mean proportional" between Template:Mvar and Template:Mvar (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">Template:Cite journal</ref> who implemented the idea as the following proof without words.<ref>Template:Cite journal</ref>
RectangleEdit
Let Template:Mvar and Template:Mvar be the points where the segments Template:Mvar and Template:Mvar intersect the semicircles Template:Mvar and Template:Mvar, respectively. The quadrilateral Template:Mvar is actually a rectangle.
- Proof: Template:Mvar, Template:Mvar, and Template:Mvar are right angles because they are inscribed in semicircles (by Thales's theorem). The quadrilateral Template:Mvar therefore has three right angles, so it is a rectangle. Q.E.D.
TangentsEdit
The line Template:Mvar is tangent to semicircle Template:Mvar at Template:Mvar and semicircle Template:Mvar at Template:Mvar.
- Proof: Since Template:Mvar is a rectangle, the diagonals Template:Mvar and Template:Mvar have equal length and bisect each other at their intersection Template:Mvar. Therefore, <math>|OD| = |OA| = |OE|</math>. Also, since Template:Mvar is perpendicular to the diameters Template:Mvar and Template:Mvar, Template:Mvar is tangent to both semicircles at the point Template:Mvar. Finally, because the two tangents to a circle from any given exterior point have equal length, it follows that the other tangents from Template:Mvar to semicircles Template:Mvar and Template:Mvar are Template:Mvar and Template:Mvar respectively.
Archimedes' circlesEdit
The altitude Template:Mvar 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 generalisationsEdit
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: "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.
EtymologyEdit
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 alsoEdit
ReferencesEdit
<references> <ref name=wolfram>Template:Mathworld</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. ("Arbelos - the Shoemaker's Knife")</ref> </references>
BibliographyEdit
- Template:Cite book
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- Template:Cite journal American Mathematical Monthly, 120 (2013), 929–935.
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