Twin circles information

In geometry, the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points $A$ , $B$ , and $C$ , and is the curvilinear triangular region between the three semicircles that have $AB$ , $BC$ , and $AC$ as their diameters. If the arbelos is partitioned into two smaller regions by a line segment through the middle point of $A$ , $B$ , and $C$ , perpendicular to line $ABC$ , 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.

These circles first appeared in the Book of Lemmas, which showed (Proposition V) that the two circles are congruent.^[1] Thābit ibn Qurra, who translated this book into Arabic, attributed it to Greek mathematician Archimedes. Based on this claim the twin circles, and several other circles in the Arbelos congruent to them, have also been called Archimedes's circles. However, this attribution has been questioned by later scholarship.^[2]

^ Cite error: The named reference archBLprop5 was invoked but never defined (see the help page).
^ Boas, Harold P. (2006). "Reflections on the Arbelos". The American Mathematical Monthly. 113 (3): 241. doi:10.1080/00029890.2006.11920301. S2CID 14528513. The source for the claim that Archimedes studied and named the arbelos is the Book of Lemmas, also known as the Liber assumptorum 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 Book of Lemmas was not Archimedes but rather some anonymous later compiler, who indeed refers to Archimedes in the third person

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[2] Boas, Harold P. (2006). "Reflections on the Arbelos". The American Mathematical Monthly. 113 (3): 241. doi:10.1080/00029890.2006.11920301. S2CID 14528513. The source for the claim that Archimedes studied and named the arbelos is the Book of Lemmas, also known as the Liber assumptorum 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 Book of Lemmas was not Archimedes but rather some anonymous later compiler, who indeed refers to Archimedes in the third person

Twin circles information

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