A circle of radius $1$ is surrounded by $4$ circles of radius $r$ as shown. What is $r$? $\textbf{(A) } \sqrt{2} \qquad$ $\textbf{(B) } 1+\sqrt{2} \qquad$ $\textbf{(C) } \sqrt{6} \qquad$ $\textbf{(D) } 3 \qquad$ $\textbf{(E) } 2+\sqrt{2}$
Solution
The centers of the four outer circles form a square of side $2r$ (neighbors are tangent), and each diagonal passes through the center of the unit circle: its length is $r+1+1+r$. So $r+1+1+r=\sqrt2 (2r)$.
$\textbf{(A) } \sqrt{2} \qquad$ $\textbf{(B) } 1+\sqrt{2} \qquad$ $\textbf{(C) } \sqrt{6} \qquad$ $\textbf{(D) } 3 \qquad$ $\textbf{(E) } 2+\sqrt{2}$ 
