# Problem #2197

 2197 Circles centered at $A$ and $B$ each have radius $2$, as shown. Point $O$ is the midpoint of $\overline{AB}$, and $OA = 2\sqrt {2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $EF$ is a common tangent. What is the area of the shaded region $ECODF$? $[asy] unitsize(6mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0); pair A=(-2*sqrt(2),0); pair B=(2*sqrt(2),0); pair G=shift(0,2)*A; pair F=shift(0,2)*B; pair C=shift(A)*scale(2)*dir(45); pair D=shift(B)*scale(2)*dir(135); pair X=A+2*dir(-60); pair Y=B+2*dir(240); path P=C--O--D--Arc(B,2,135,90)--G--Arc(A,2,90,45)--cycle; fill(P,gray); draw(Circle(A,2)); draw(Circle(B,2)); dot(A); label("A",A,W); dot(B); label("B",B,E); dot(C); label("C",C,W); dot(D); label("D",D,E); dot(G); label("E",G,N); dot(F); label("F",F,N); dot(O); label("O",O,S); draw(G--F); draw(C--O--D); draw(A--B); draw(A--X); draw(B--Y); label("2",midpoint(A--X),SW); label("2",midpoint(B--Y),SE); [/asy]$ $\text{(A)}\ \frac {8\sqrt {2}}{3} \qquad \text{(B)}\ 8\sqrt {2} - 4 - \pi \qquad \text{(C)}\ 4\sqrt {2} \qquad \text{(D)}\ 4\sqrt {2} + \frac {\pi}{8} \qquad \text{(E)}\ 8\sqrt {2} - 2 - \frac {\pi}{2}$ This problem is copyrighted by the American Mathematics Competitions.
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• Reduce fractions to lowest terms and enter in the form 7/9.
• Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
• Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
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