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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
323:
422:
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divides the arbelos into two regions, each bounded by a semicircle, a straight line segment, and an arc of the outer semicircle. The circles
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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
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154:, and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles (with diameters
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1130:(reprint of 1929 edition by Houghton Mifflin ed.). New York: Dover Publications. pp. 116–117.
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the circle on PN as diameter, where PN is perpendicular to AB and meets the original semicircle in P.
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Advanced
Euclidean Geometry: An elementary treatise on the geometry of the triangle and the circle
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segments instead of half circles. A generalisation comprising both arbelos and parbelos is the
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478:(Euclid, Book VI, Proposition 8, Porism). This proof approximates the ancient Greek argument;
76:, where some of its mathematical properties are stated as Propositions 4 through 8. The word
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from antiquity to the current day, whose blade is said to resemble the geometric figure.
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is a right angle. The sum of the angles in any quadrilateral is 2π, so in quadrilateral
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172:). Since the area of a circle is proportional to the square of the diameter (
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Two of the semicircles are necessarily concave, with arbitrary diameters
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954:
864:-belos, which uses a certain type of similar differentiable functions.
80:
is Greek for 'shoemaker's knife'. The figure is closely related to the
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Sondow, J. (2013). "The parbelos, a parabolic analog of the arbelos".
146:: For the proof, reflect the arbelos over the line through the points
1084:
173:
1040:
450:, being inscribed in the semicircle, has a right angle at the point
1027:
Nelsen, R B (2002). "Proof without words: The area of an arbelos".
840:
19:
1187:
886:
839:
349:, so this equation simplifies algebraically to the statement that
166:) are subtracted from the area of the large circle (with diameter
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26:
18:
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are right angles because they are inscribed in semicircles (by
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The type of shoemaker's knife that gave its name to the figure
134:
of the arbelos is equal to the area of a circle with diameter
492:
180:, Book XII, Proposition 2; we do not need to know that the
1220:
The
Penguin Dictionary of Curious and Interesting Geometry
571:
therefore has three right angles, so it is a rectangle.
454:(Euclid, Book III, Proposition 31), and consequently
355:
210:
1004:. Cambridge University Press. Proposition 4 in the
318:{\displaystyle 2|AH|^{2}=|BC|^{2}-|AC|^{2}-|BA|^{2}}
760:(the rectangle's diagonal) is also the midpoint of
424:. Thus the claim is that the length of the segment
1217:
416:
317:
66:The earliest known reference to this figure is in
911:, meaning "shoemaker's knife", a knife used by
856:is a figure similar to the arbelos, that uses
8:
486:who implemented the idea as the following
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16:Plane region bounded by three semicircles
462:is indeed a "mean proportional" between
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828:in each of these regions, known as the
670:is a right angle). Therefore triangles
204:), the problem reduces to showing that
766:(the rectangle's other diagonal). As
7:
1224:. New York: Penguin Books. pp.
1197:10.4169/amer.math.monthly.120.10.929
832:of the arbelos, have the same size.
121:Some special points on the arbelos.
47:is a plane region bounded by three
14:
1072:The American Mathematical Monthly
1015:"Arbelos - the Shoemaker's Knife"
513:be the points where the segments
417:{\displaystyle |AH|^{2}=|BA||AC|}
1261:
1249:
1113:, Volume 13 (2013), pp. 103–111.
750:is a rectangle, so the midpoint
446:. Now (see Figure) the triangle
434:of the lengths of the segments
836:Variations and generalisations
776:) is the center of semicircle
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333:equals the sum of the lengths
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1:
1266:The dictionary definition of
1210:American Mathematical Monthly
1067:"Reflections on the Arbelos"
1000:Thomas Little Heath (1897),
770:(defined as the midpoint of
746:must be a right angle. But
182:constant of proportionality
1302:
525:intersect the semicircles
100:; the third semicircle is
1105:Antonio M. Oller-Marcen:
869:Poincaré half-plane model
800:is tangent to semicircle
796:. By analogous reasoning
788:is tangent to semicircle
587:is tangent to semicircle
23:An arbelos (grey region)
1126:Johnson, R. A. (1960).
1002:The Works of Archimedes
925:Archimedes' quadruplets
875:, an arbelos models an
784:is a right angle, then
726:. Therefore the sum of
714:is a straight line, so
1212:, 120 (2013), 929–935.
1151:Excursions in Geometry
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59:) that contains their
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567:). The quadrilateral
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31:Arbelos sculpture in
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1258:at Wikimedia Commons
724:supplementary angles
353:
208:
1175:Amer. Math. Monthly
1111:Forum Geometricorum
830:Archimedes' circles
816:Archimedes' circles
704:is the midpoint of
694:is the midpoint of
488:proof without words
1216:Wells, D. (1991).
1154:. Dover. pp.
978:Weisstein, Eric W.
893:
850:
614:is a right angle,
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1254:Media related to
1137:978-0-486-46237-0
482:cites a paper of
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1181:(10): 929–935.
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595:and semicircle
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1244:External links
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1079:(3): 236–249.
1054:
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1006:Book of Lemmas
993:
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952:
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937:
935:Schoch circles
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930:Bankoff circle
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884:
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877:ideal triangle
844:example of an
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813:
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540:is actually a
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480:Harold P. Boas
432:geometric mean
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73:Book of Lemmas
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1272:at Wiktionary
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1146:Ogilvy, C. S.
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1119:
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1108:
1107:"The f-belos"
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831:
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820:The altitude
811:
725:
681:
609:
606:
605:
604:
578:
574:
566:
550:
547:
546:
545:
543:
536:
535:quadrilateral
500:
495:
491:
489:
485:
481:
433:
406:
403:
390:
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379:
374:
364:
361:
325:. The length
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53:straight line
50:
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35:, Netherlands
34:
29:
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1268:
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1127:
1120:Bibliography
1110:
1101:
1076:
1070:
1057:
1032:
1028:
1022:
1009:
1005:
1001:
996:
984:
950:Pappus chain
908:
904:
896:
894:
866:
861:
851:
845:
819:
809:
780:, and angle
682:. Therefore
644:also equals
607:
582:
572:
548:
504:
143:
142:
129:
110:
106:
91:
82:Pappus chain
77:
71:
65:
56:
44:
38:
945:Woo circles
940:Schoch line
907:or ἄρβυλος
899:comes from
640:. However,
49:semicircles
33:Kaatsheuvel
1035:(2): 144.
1008:. Quote:
961:References
905:he árbēlos
903:ἡ ἄρβηλος
88:Properties
68:Archimedes
1188:1210.2279
1029:Math. Mag
986:MathWorld
981:"Arbelos"
895:The name
883:Etymology
826:inscribed
583:The line
542:rectangle
501:Rectangle
290:−
264:−
61:diameters
1280:Category
1205:33402874
1148:(1990).
1093:27641891
1065:(2006).
919:See also
913:cobblers
858:parabola
854:parbelos
690:, where
610:: Since
579:Tangents
178:Elements
57:baseline
41:geometry
1286:Arbelos
1269:arbelos
1256:Arbelos
1049:3219152
955:Salinon
909:árbylos
897:arbelos
871:of the
867:In the
686:equals
680:similar
666:(since
659:
647:
633:
621:
618:equals
430:is the
201:
187:
78:arbelos
45:arbelos
1232:
1203:
1162:
1134:
1109:. In:
1091:
1047:
848:-belos
810:Q.E.D.
734:is π.
710:. But
662:minus
636:minus
573:Q.E.D.
559:, and
476:|
472:|
468:|
464:|
460:|
456:|
347:|
343:|
339:|
335:|
331:|
327:|
174:Euclid
102:convex
1201:S2CID
1183:arXiv
1156:51–54
1089:JSTOR
1045:JSTOR
901:Greek
608:Proof
549:Proof
144:Proof
55:(the
43:, an
1230:ISBN
1160:ISBN
1132:ISBN
852:The
782:∠IDE
748:ADHE
744:∠IDO
740:IDOA
736:∠IAO
732:∠DOA
730:and
728:∠DIA
722:are
720:∠DOA
718:and
716:∠DOH
712:∠AOH
700:and
688:∠DOH
684:∠DIA
678:are
674:and
668:∠HAB
664:∠DAB
642:∠DAH
638:∠DAB
616:∠DBA
612:∠BDA
569:ADHE
561:∠AEC
557:∠BHC
553:∠BDA
538:ADHE
529:and
519:and
509:and
505:Let
470:and
440:and
341:and
150:and
132:area
130:The
126:Area
96:and
1226:5–6
1193:doi
1179:120
1081:doi
1077:113
1037:doi
804:at
792:at
754:of
676:DAH
672:DBA
599:at
591:at
448:BHC
184:is
176:'s
70:'s
39:In
1282::
1228:.
1199:.
1191:.
1177:.
1158:.
1087:.
1075:.
1069:.
1043:.
1033:75
1031:.
983:.
968:^
879:.
822:AH
808:.
802:AC
798:DE
790:BA
786:DE
778:BA
773:BA
763:DE
757:AH
742:,
707:AH
697:BA
603:.
597:AC
589:BA
585:DE
555:,
551::
544:.
531:AC
527:AB
522:CH
516:BH
490:.
474:AC
466:BA
458:HA
443:AC
437:BA
427:AH
345:AC
337:BA
329:BC
169:BC
163:AC
160:,
157:BA
140:.
137:HA
84:.
63:.
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1013:(
989:.
862:f
846:f
806:E
794:D
768:I
752:O
702:O
692:I
656:2
653:/
650:π
630:2
627:/
624:π
601:E
593:D
511:E
507:D
452:H
411:|
407:C
404:A
400:|
395:|
391:A
388:B
384:|
380:=
375:2
370:|
365:H
362:A
358:|
311:2
306:|
301:A
298:B
294:|
285:2
280:|
275:C
272:A
268:|
259:2
254:|
249:C
246:B
242:|
238:=
233:2
228:|
223:H
220:A
216:|
212:2
198:4
195:/
191:π
152:C
148:B
113:.
111:b
109:+
107:a
98:b
94:a
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