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Find radius of a tangent circle to another circle with the center on axis

I have one circle C1, its center and radius are known. And a segment AB where the points A and B are known and alwyas situated in the circle C1.
The segment AB is not a chord of the circle C1.
How to obtain the radius of the circle C2 if its center is located on the segment AB,
AND the circle C2 is tangent to the circle C1,
AND the circle C2 passing by the B point ?

I suppose there is only one solution of circle for each couple C1 and segment AB.

http://sofiasi.net/data/Yannick/tangent_circle.PNG

http://sofiasi.net/data/Yannick/tangent_circle_explanations.PNG

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