AMC 10 2014b Problem 22

Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles? $\text{(A) } \dfrac{1+\sqrt2}4 \text{(B) } \dfrac{\sqrt5-1}2 \text{(C) } \dfrac{\sqrt3+1}4 \text{(D) } \dfrac{2\sqrt3}5 \text{(E) } \dfrac{\sqrt5}3$