In the Figure, you have three circles: Large, middle size and small circle A.


        Let "a" be the radius of the small circle A, which is under the question.


    Draw vertical radius of the Large circle from its center.

    Connect the centers of the small and the middle size circles.

    Draw the horizontal segment from the center of the small circle till the intersection with the vertical radius of the Large circle.

    Let H be the length of this horizontal segment.

    You get right-angled triangle with the legs H, 2-a and the hypotenuse 2+a.

    Hence, you have a Pythagorean equation

         =  + .     (1)


    Next, draw the radius of the Large circle from its center through the center of the small circle A.

    Draw the vertical segment from the center of the small circle A to the horizontal diameter of the Large circle.

    You will get right-angled triangle with the legs "a", H and the hypotenuse (2-a).

    Hence, you have second Pythagorean equation

         =  + .         (2)


    OK. The setup is completed, and you have now two equations for two unknowns


         =  + .     (1)

         =  + .         (2)


    Subtract eq(2) from eq(1) to eliminate H.  You will get a single equation for "a"

         = ,

        2a*2 =   ====>

        4a = 4 - 4a  ====>  4a + 4a = 4  ====>  8a = 4  ====>  a =  = .


Answer.  The radius of the small circle is  a = .