Archimedes' quadruplets

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

In geometry, Archimedes' quadruplets, introduced by Frank Power in the summer of 1998, each have the same area as Archimedes' twin circles.^[1]

[edit] Construction

Three semicircles are created with the diameters of AB, AC, and BC. The two smaller circles have radii of r₁ and r₂, meaning the larger semicircle has a radius that satisfies the equation: r = r₁+r₂. Let the points D and E be the center and midpoint, respectively, of the semicircle with the radius r₁. Let H be the midpoint of line AC.

[edit] Proof of congruency

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

$\frac{r_1\cdot r_2}{r}.$

By the Pythagorean theorem:

$\left(HE\right)^2=\left(r_1\right)^2+\left(r_2\right)^2.$

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

$\left(HJ_i\right)^2=\left(HE\right)^2+x^2=\left(r_1\right)^2+\left(r_2\right)^2+x^2$