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.
Read more about this topic: Archimedes' Quadruplets
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