Archimedes' Quadruplets - Proof of Congruency

Proof of Congruency

According to Proposition 5 of Archimedes' Book of Lemmas, the common radius of Archimedes' twin circles is:

By the Pythagorean theorem:

Then, create two circles with centers Ji perpendicular to HE, tangent to the large semicircle at point Li, tangent to point E, and with equal radii x. Using the Pythagorean theorem:

Also:

Combinding these gives:

Expanding, collecting to one side, and factoring:

Solving for x:

Proving that each of the Archimedes' quadruplets' areas is equal to each of Archimedes' twin circles' areas.

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