Starting with the worst case: the small circle skew in all axes, at an arbitrary location, and for the sake of argument, assuming it's not quicker just to create another, centered, small circle.
Pivot set to '3D Cursor', Snap set to 'Vertex' and 'Active'
To center the small circle to the large one
- Select all vertices of the large circle
- ShiftS > Cursor to Selected
- Select all vertices of the small circle
- ShiftS > Selected to Cursor (Offset)
To orient the small circle to the large one
Having centered the small circle, with the 3D cursor also at the center
- Select all vertices of the small circle
- With one of its vertices active, RX, RY,RZ in turn, snapping the rotations to any of the vertices of the large circle
To slide the small circle along a bisector of the large one
Having centered and aligned the small circle
- Select the two vertices at the ends of your bisector of choice.
- Hit CtrlAltSpace to make a Custom Orientation from them, (with 'Use After Creation' checked)
- Select all the vertices of the small circle, and GYY move it along your new orientation.