I hope this doesn't come too late...

I assume that you have a circular sector.

1. Construct the angle bisector passing through S. The intersection of the angle bisector and the arc of the sector is Z. Z must be a tangent point of the circle in question.

2. Choose an arbitrary point on the angle bisector which is the midpoint of the arbitrary circle with the radius (in blue)

3. Draw a line through perpendicular on one leg of the sector. This line intersects the circle in

4. Draw a line which intersect the leg of the sector in T. This is a tangentpoint of the circle c which you are looking for.

5. A parallel to through T intersects the angle bisector in M which is the midpoint of the circle c. (in red)

Actually the construction uses similar triangles (indicated by different patterns in red) and the point Z as the fixpoint of a central dilation.