No such thing as a silly question. I don't think you need a special gadget, provided that the board is big enough for there to be a gap between the base of your router as it cuts the groove and the little 'island' of wood temporarily attached to the middle. If you make the island the same thickness as your router base, all you need is a stick to bridge across the two things, held on with screws through the router base and one screw in the middle as a pivot point. You'll need to find the exact centre.
However, now that you have cut a circle, I think it would be easier to use the router fence to reference off the outer edge, which you can do with the ordinary fence, provided that you put a couple of blocks onto it with a gap between, so there are always two points of contact against the circle.
Then again you could always use a Stanley 71 with the fence in the position for circular work, ie with the two bumps inwards - that's what I would do!