Circles (A_1a) and (A_1b): The Archimedean twins

The Adam and Eve of Archimedean circles in the arbelos.

Through C draw the perpendicular to AB, dividing the arbelos in two parts. The incircles of these parts are congruent. [Archimedes]

Properties

• The circular hull of the Archimedean twins has the same area as the arbelos itself (and the circle with diameter CD). [Bankoff 1954]
• The common tangent of (A_1a) with  (O₁) passes through B, the common tangent of (A_1b) and (O₂) passes through A. [Dodge e.a. 2000]
• The common tangent segment AT_b is congruent to AD, the common tangent segment BT_a is congruent to BD. [Dodge e.a. 2000]
• Let N₁ be the point where O₁A_1a meets CD then O₁N₁ = O₁B. Let N₂ be the oint where O₂A_1b meets CD, then O₂N₂ = O₂B.[Wendijk 2001/Hermann 2003, d'Ignazio and Suppa 2001 - reworded]

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