Exercise:

The most northerly star of the Southern Cross, γ Crucis, has declination -57°.
At what latitude will it just be visible?

The star is at S (just on the horizon), 57° from the equator.
So at this place, it must be 33° from the equator to the zenith.
So it must be 57° from the zenith to the north celestial pole.
So it must be 33° from the pole to the northern horizon.
But the altitude of the north celestial pole is the latitude of the place.
So the latitude is 33°N.
So any observer north of latitude 33°N is unable to see the Southern Cross.

At what latitude will it pass directly overhead?

The star is at Z, the zenith.
It is 57° from there to the equator,
so at this place it must be 33° from the equator to the horizon.
So P is 57° below the northern horizon.
So the latitude is 57°S.
Note: as a general rule, if a star of declination x° passes overhead,
then the place has latitude x°.

At what latitudes will it never set?

Suppose the star is at S, just on the southern horizon.
It is 57° from S down to the equator
so it must be 33° from S up to the south celestial pole.
If the SCP is 33° above the southern horizon,
then the NCP must be 33° below the northern horizon.
So the latitude here is -33°, or 33°S.
The star will never set (it will be circumpolar)
for any observer south of 33°S.

Back to "the HA-dec system".