Problem
2019 AMC 10B Problem 20
As shown in the figure, line segment \overline{AD} is trisected by points B and C so that AB=BC=CD=2. Three semicircles of radius 1, \text{arc }AEB,\text{arc }BFC, and \text{arc }CGD, have their diameters on \overline{AD}, lie in the same halfplane determined by line AD, and are tangent to line EG at E,F, and G, respectively. A circle of radius 2 has its center on F. The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form \frac{a}{b}\cdot\pi-\sqrt{c}+d,where a,b,c, and d are positive integers and a and b are relatively prime. What is a+b+c+d?
![[asy] size(6cm); filldraw(circle((0,0),2), gray(0.7)); filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0)); dot((-3,-1)); label("$A$",(-3,-1),S); dot((-2,0)); label("$E$",(-2,0),NW); dot((-1,-1)); label("$B$",(-1,-1),S); dot((0,0)); label("$F$",(0,0),N); dot((1,-1)); label("$C$",(1,-1), S); dot((2,0)); label("$G$", (2,0),NE); dot((3,-1)); label("$D$", (3,-1), S); [/asy]](/static/aops/be5538c17cf2af3d37027f4996d5a8ae5cd4aef1.png)
\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16\qquad\textbf{(E) } 17
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