Another Pair of Twins in Arbelos: What is this about?
A Mathematical Droodle

Explanation

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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...

• Arbelos - the Shoemaker's Knife
• 7 = 2 + 5 Sangaku
• Another Pair of Twins in Arbelos
• Archimedes' Quadruplets
• Archimedes' Twin Circles and a Brother
• Book of Lemmas: Proposition 5
• Book of Lemmas: Proposition 6
• Chain of Inscribed Circles
• Concurrency in Arbelos
• Concyclic Points in Arbelos
• Ellipse in Arbelos
• Gothic Arc
• Pappus Sangaku
• Rectangle in Arbelos
• Squares in Arbelos
• The Area of Arbelos
• Twin Segments in Arbelos
• Two Arbelos, Two Chains
• A Newly Born Pair of Siblings to Archimedes' Twins

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They were two of a set of triplets or quadraplets or ...

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