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Explanation

In Proposition 5 of the Book of Lemmas Archimedes found two equal circles, known as Archimedes' twins, on both sides of the segment of the line perpendicular to the common base of the semicircles forming an arbelos from the point of tangency of the two small semicircles to the point where it crosses the big one.

In Proposition 4, he found a circle whose area equals that of the arbelos. The statement is simple and admits a proof without words. The circle in question has as a diameter the aforementioned line segment. Let's call this circle a big brother of the twins. The family is actually very big. Until a few years ago, the twins were thought to belong to a triplet of equal circles. Eventually, more than 30, I believe, of their equal siblings have been discovered.

Interestingly, the big brother also has a twin. The smallest circle that encloses Archimedes' twins is equal to the big brother in all respects.

To be continued...