A Sangaku Problem

Consider a circle O with a diameter AB. Draw a second circle with diameter AC, such that C is on AB in circle O. Draw an isosceles triangle with base CB and third vertex on circle O. Draw a third circle (X), tangent to the first three figures. Prove that the line from C to the center of circle X is perpendicular to AB.
Source: Based upon puzzle 2 in Ivan Moscovich's 1000 PlayThinks, though I've found it elsewhere. A good Scientific American article on sangaku problems: http://www.sciam.com/1998/0598issue/0598rothman.html
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