Problem
1957 AHSME Problem 47
In circle O, the midpoint of radius OX is Q; at Q, \overline{AB} \perp \overline{XY}. The semi-circle with \overline{AB} as diameter intersects \overline{XY} in M. Line \overline{AM} intersects circle O in C, and line \overline{BM} intersects circle O in D. Line \overline{AD} is drawn. Then, if the radius of circle O is r, AD is:
![[asy] defaultpen(linewidth(.8pt)); unitsize(2.5cm); real m = 0; real b = 0; pair O = origin; pair X = (-1,0); pair Y = (1,0); pair Q = midpoint(O--X); pair A = (Q.x, -1*sqrt(3)/2); pair B = (Q.x, -1*A.y); pair M = (Q.x + sqrt(3)/2,0); m = (B.y - M.y)/(B.x - M.x); b = (B.y - m*B.x); pair D = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b)); m = (A.y - M.y)/(A.x - M.x); b = (A.y - m*A.x); pair C = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b)); draw(Circle(O,1)); draw(Arc(Q,sqrt(3)/2,-90,90)); draw(A--B); draw(X--Y); draw(B--D); draw(A--C); draw(A--D); dot(O);dot(M); label("$B$",B,NW); label("$C$",C,NE); label("$Y$",Y,E); label("$D$",D,SE); label("$A$",A,SW); label("$X$",X,W); label("$Q$",Q,SW); label("$O$",O,SW); label("$M$",M,NE+2N);[/asy]](/static/aops/d40adb845b830ab1e2f4f3b2600588d2140efaf7.png)
\textbf{(A)}\ r\sqrt{2}\qquad\textbf{(B)}\ r\qquad\textbf{(C)}\ \text{not a side of an inscribed regular polygon}\qquad\textbf{(D)}\ \frac{r\sqrt{3}}{2}\qquad\textbf{(E)}\ r\sqrt{3}
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