Archimedes' quadruplets

In geometry, Archimedes' quadruplets are four congruent circles associated with an arbelos.

[1][2][3] An arbelos is formed from three collinear points A, B, and C, by the three semicircles with diameters AB, AC, and BC.

The other two quadruplet circles are formed in a symmetric way from the semicircle with radius r2.

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: Combining 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.

Each of the Archimedes' quadruplets (green) have equal area to each other and to Archimedes' twin circles