Some Generalizations

  1. ( The twins are triplets) In Figure 5, let $P$ and $Q$ be the points where the circle inscribed in the arbelos is tangent to the two smaller semicircles, and let $R$ be the point where the two smaller semicircles are tangent. The circle containing $P$, $Q$ and $R$ has a radius equal to that of the twin circles, namely $r_1r_2/r$. See [ 8].

  2. ( Quadruplets) The largest circle inscribed in the circular segment of the largest semi- circle formed by the chord $\overline{EF}$ tangent to the other two semicircles also has the same radius as the twins. See Figure 6. Note that the point of tangency for this circle is the point $D$ of the common internal tangent to the smaller semicircles of the arbelos, $\overline {CD}$. The story doesn't end here. There are infinite families of circles in and around the arbelos with the same radius as the twins. See [ 10] and [ 11].
  3. In Figure 7, a chain of circles, inscribed in two internally tangent semicircles with collinear endpoints, begins with a circle tangent to the line of centers of the semicircles. The distance of the center of the $n$th circle in the chain from the line of centers is $(2n-1)r_n$ where $r_n$ is the radius of the $n$th circle.

  4. If the radius of the largest semicircle in an arbelos is $k$ times the radius of the smallest semicircle, where $k$ is an integer, then at least one pair of circles in the chain will have their centers on a line parallel to the base of the arbelos. The circles paired in this manner are those whose order numbers in the chain have a product equal to $k(k-1)$. See On a Generalization of the Arbelos by M. G. Gaba in the American Mathematical Monthly, vol 47, Jan. 1940, pp 19-24.

  5. ( Valentine) Let $r,r_1,r_2$ be the radii of the semicircles forming an arbelos and let $\rho$ be the radius of the inscribed circle. In Figure 8, prove the ratio of the area of the arbelos to the area of the heart equals $\displaystyle \frac{\rho}{r}=\frac{r^2-r_1^{\;2}-r_2^{\;2}}{r^2+r_1^{\;2}+r_2^{\;2}}$. See [ 9] for ten proofs by Charles W. Trigg. See [ 4] for seven proofs that $\displaystyle \rho= \frac{rr_1r_2}{r_1^{\;2}+r_1r_2+r_2^{\;2}}$.