right|thumb|320px|An arbelos (grey region)
thumb|Arbelos sculpture in [[Kaatsheuvel, Netherlands]]
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. who implemented the idea as the following proof without words.
center
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 rectangle, the diagonals and have equal length and bisect each other at their intersection . Therefore, <math>|OD| = |OA| = |OE|</math>. Also, since is perpendicular to the diameters and , is tangent to both semicircles at the point . Finally, because the two tangents to a circle from any given exterior point have equal length, it follows that the other tangents from to semicircles and are and respectively.
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
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.
In the Poincaré half-plane model of the hyperbolic plane, an arbelos models an ideal triangle.
Etymology
right|thumb|upright=1.0|The type of shoemaker's knife that gave its name to the figure
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
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