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Let AB be the diameter of a semicircle, C any point on AB, and CD perpendicular to it, and let semicircles be described within the first semicircle and having AC, CB as diameters. Then if two circles be drawn touching CD on different sides and each touching two of the semicircles, the circles so
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from the title of the seventeenth century Latin translation of the ninth-century Arabic translation of the lost Greek original. Although this collection of fifteen propositions is included in standard editions of the works of
Archimedes, the editors acknowledge that the author of the
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Alternative approaches to constructing two circles congruent to the twin circles have also been found. These circles have also been called
Archimedean circles. They include the
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91:, then each of the two twin circles lies within one of these two regions, tangent to its two semicircular sides and to the splitting segment.
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116:. Based on this claim the twin circles, and several other circles in the Arbelos congruent to them, have also been called
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as their diameters. If the arbelos is partitioned into two smaller regions by a line segment through the middle point of
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Each of the two circles is uniquely determined by its three tangencies. Constructing it is a special case of the
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was not
Archimedes but rather some anonymous later compiler, who indeed refers to Archimedes in the third person
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divides the arbelos in two parts. The twin circles are the two circles inscribed in these parts, each
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Alternatively, if the outer semicircle has unit diameter, and the inner circles have diameters
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The smallest circle that encloses both twin circles has the same area as the arbelos.
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be the diameters of two inner semicircles, so that the outer semicircle has diameter
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The source for the claim that
Archimedes studied and named the arbelos is the
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Animation of twin circles for various positions of point B on AC segment
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120:. However, this attribution has been questioned by later scholarship.
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690:""Archimedes' Circles." From MathWorld—A Wolfram Web Resource"
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be the point where the larger semicircle intercepts the line
59:, and is the curvilinear triangular region between the three
108:, who translated this book into Arabic, attributed it to
100:, which showed (Proposition V) that the two circles are
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to one of the two smaller semicircles, to the segment
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47:. An arbelos is determined by three collinear points
602:. Cambridge University Press. Proposition 5 in the
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419:. The diameter of each twin circle is then
711:A catalog of over fifty Archimedean circles
43:are two special circles associated with an
31:The twin circles (red) of an arbelos (grey)
196:be the three corners of the arbelos, with
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731:Online document, accessed on 2014-10-08.
715:Online document, accessed on 2014-10-08.
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520:, the diameter of each twin circle is
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464:{\displaystyle d={\frac {ab}{a+b}}.}
94:These circles first appeared in the
727:Circles (A61a) and (A61b): Dao pair
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631:The American Mathematical Monthly
373:, and to the largest semicircle.
644:10.1080/00029890.2006.11920301
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539:
1:
18:Archimedes' twin circles
626:"Reflections on the Arbelos"
598:Thomas Little Heath (1897),
561:{\displaystyle d=s(1-s).\,}
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724:Floor van Lamoen (2014),
708:Floor van Lamoen (2014),
87:, perpendicular to line
600:The Works of Archimedes
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118:Archimedes's circles
688:Weisstein, Eric W.
513:{\displaystyle 1-s}
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366:{\displaystyle BD}
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339:{\displaystyle BD}
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303:through the point
296:{\displaystyle AC}
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136:Specifically, let
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662:Liber assumptorum
487:{\displaystyle s}
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316:{\displaystyle B}
269:{\displaystyle D}
249:{\displaystyle C}
229:{\displaystyle A}
209:{\displaystyle B}
189:{\displaystyle C}
169:{\displaystyle B}
149:{\displaystyle A}
16:(Redirected from
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323:. The segment
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667:Book of Lemmas
658:Book of Lemmas
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604:Book of Lemmas
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389:Schoch circles
385:Bankoff circle
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97:Book of Lemmas
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278:perpendicular
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693:. Retrieved
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124:Construction
117:
95:
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41:twin circles
40:
34:
606:. Quote: "
581:Schoch line
393:Woo circles
61:semicircles
695:2008-04-10
638:(3): 241.
587:References
399:Properties
114:Archimedes
63:that have
546:−
505:−
102:congruent
741:Category
652:14528513
624:(2006).
575:See also
216:between
37:geometry
747:Arbelos
348:tangent
280:to the
256:. Let
45:arbelos
650:
391:, and
176:, and
83:, and
71:, and
55:, and
39:, the
648:S2CID
110:Greek
494:and
407:and
403:Let
236:and
640:doi
636:113
89:ABC
35:In
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676:^
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634:.
628:.
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156:,
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73:AC
69:BC
67:,
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51:,
729:.
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534:=
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453:b
450:+
447:a
442:b
439:a
433:=
430:d
417:b
413:a
409:b
405:a
361:D
358:B
334:D
331:B
311:B
291:C
288:A
264:D
244:C
224:A
204:B
184:C
164:B
144:A
85:C
81:B
77:A
57:C
53:B
49:A
20:)
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