Problem
2005 AMC 8 Problem 23
Isosceles right triangle ABC encloses a semicircle of area 2\pi. The circle has its center O on hypotenuse \overline{AB} and is tangent to sides \overline{AC} and \overline{BC}. What is the area of triangle ABC?
![[asy]pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2); draw(circle(o, 2)); clip(a--b--c--cycle); draw(a--b--c--cycle); dot(o); label("$C$", c, NW); label("$A$", a, NE); label("$B$", b, SW);[/asy]](/static/aops/8a7fba91b0cdfee41e69cbd89d6b1389d4169877.png)
\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 3\pi\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 4\pi
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