Arctic Circle being an imaginary circle round the globe at about 66° 34' where there is no sunset in the summer and no sunrise in the winter.
Not entirely true. Some few metres north of the polar circle, there will be one summer night where the lower edge of the sun just touches the horizon before rising again, and one winter day where the sun stays below the horizon even at the middle of the day. The rest of the year there will be days and nights.
Just like I also mentioned above, that was one of the given definition of the Arctic by a scientist, from what you said it seem to have a flaw, it have been seen to be given other definitions though, of which I stated some above, thanks for pointing the flaw in that particular definition :)