Given circles $c$ and $d$ (green circles) and chord $AB$ arranged as shown in the figure below. How can I construct the blue circle tangent to all previous elements?
[Once I think I found an elementary construction which unfortunately I cannot remember.]
Hint: Let small green circle touch big one and a chord at $M$ and $N$. Then it is not difficult to see that line $MN$ cuts small arc $AB$ at it midpoint, say $S$ (check it by, say homothety). From $S$ draw a tangent to small green and let it cut $AB$ at $P$ and touch small green at $D$. Now draw a angle bisector for angle $\angle BPD$. Blue circle has center on it and touch a line $SD$ at $D$. (All these can be confirmed by radical axsis and/or inversion at $S$.)