Archimedes' twin circles

In geometry, specifically in the study of the arbelos, the twin circles are two special circles associated with it. 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.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' circles. However, this attribution has been questioned by later scholarship.

Specifically, let ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ be the three corners of the arbelos, with ${\displaystyle B}$ between ${\displaystyle A}$ and ${\displaystyle C}$. Let ${\displaystyle H}$ be the point where the larger semicircle intercepts the line perpendicular to the ${\displaystyle AC}$ through the point ${\displaystyle B}$. The segment ${\displaystyle BH}$ divides the arbelos in two parts. The twin circles are the two circles inscribed in these parts, each tangent to one of the two smaller semicircles, to the segment ${\displaystyle BH}$, and to the largest semicircle.

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