As mentioned by Barry in the comments, you just have to substract 6 degrees of latitude to get the ring around areas were it never gets DARKER than civil twilight or darker for 24 or more hours (or 12° for nautical twilight and 18° for astronomical twilight). That would be:
- "Never darker than civil twilight circle": : 60° 33′ 46.7″ (orange in the figures below)
- "Never darker than nautical twilight circle": : 54° 33′ 46.7″ (green in the figures below)
- "Never darker than astronomical twilight circle": : 48° 33′ 46.7″ (light blue in the figures below)
To be clear, those circles surround the places where on summer solstice there is no complete nightfall (i.e. it doesn't get completely dark according to the different twilight standards).
In turn, you can think about other set of equivalent circles where it never get BRIGHTER than a given type of twilight for at least 24 hours, those would be
- "Never brighter than civil twilight circle": : 72° 33′ 46.7″ (orange dotted)
- "Never brighter than nautical twilight circle": : 78° 33′ 46.7″ (green dotted)
- "Never brighter than astronomical twilight circle": : 84° 33′ 46.7″ (light blue dotted)
To be clear again, those circles surround the places where on winter solstice there is no tace of daylight (i.e. they doesn't get any daylight according to the different twilight standards).
In a map, these 6 circles, plus the Polar circle (at latitude 66° 33′ 46.7″ and in red) would look like this around the North pole
And like this around the South Pole
And to have a world view, in geographic projection, they would look like this
The twilights are defined like that (i.e. Sun 6, 12, 18° below the horizon), so no corrections have to be made due to refraction or the size of the solar disk. Such corrections would only apply if you want to find the "effective" polar circle where the Sun would/would not be visible for 24 hours even with an horizon perfectly free of obstacles.
Due to those corrections, the polar night (at the Winter Solstice) only really happens some 80-100 km poleward of the polar circle. In contrast, at the polar circle itself, the Sun can be seen even at the winter solstice.