A semicircle has diameter $\overline{AB}$. Two congruent smaller semicircles have their diameters on $\overline{AB}$; they are tangent to each other and each is internally tangent to the large semicircle (so their diameters together fill $\overline{AB}$). A chord $\overline{CD}$ of the large semicircle, of length $20$, is parallel to $\overline{AB}$ and tangent to both small semicircles. The two small semicircles are cut away from the large one. What is the area of the resulting figure?

1. Let each small semicircle have radius $\rho$. Their diameters fill $\overline{AB}$, so the large semicircle has radius $R=2\rho$, and its center $O$ is the point where the small semicircles touch.
2. The centers of the small semicircles are at distance $\rho$ from $O$ on $\overline{AB}$, so a line tangent to the tops of both small semicircles is at height $\rho$ above $\overline{AB}$. Thus $\overline{CD}$ is at height $\rho$, and its half-length is $\sqrt{R^2-\rho^2}=\sqrt3\,\rho=10$, so $\rho^2=\frac{100}3$.
3. The area is $\frac12\pi(2\rho)^2-2\cdot\frac12\pi\rho^2=\pi\rho^2=\boxed{\dfrac{100\pi}3}$.