I came up with Sean’s solution and assumed that was what Jeff had in mind. I completely missed Joe’s.

The point of Sean’s solution is symmetry. The original circle and the circle centred on the hole are identical but off-set by two inches. They overlap each other in a symmetrical way. The two areas which are in one circle, but not the other, are identical crescent shapes. Adding either crescent to the overlapping region will give one of the two circles.

I thought this was very neat!

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