I am currently self teaching myself some basic astronomy. My book claims, if a constellation is observed for exactly 6 months during a year, then it lies on the celestial equator. I have failed to understand why this must be true. Shouldn't the visible duration depends on the position of the observer? For example, for an observer at equator, wouldn't all constellations (accept those at the polaris) be visible for exactly half a year? Also, is the constellation at the NCP always visible for people in the northern hemisphere? Thank you in advance!
Taken at face value it's not correct that points on the equator are visible for exactly six months.
For example, Orion is on the celestial equator. It is true that Orion will be above the horizon for half the day (12 hours), but during the summer when it is above the horizon, so is the sun, so it can't be seen.
But Orion is visible from Northern Europe between the middle of August (when it rises shortly before the sun) and the middle of April (when it sets shortly after dusk) That is rather more than six months. Exact dates can't be given as visibility during twilight depends on factors like the clarity of the atmosphere, and stars don't wink off as the sun rises, they fade out.
On the other hand, if you always observe at the same time then points on the celestial equator will be above the horizon for six months of the year, but just being above the horizon doesn't make it visible, when close to the horizon the thicker atmosphere (and more prosaically, trees and buildings) will make a constellation effectively obscured.
It is true that the north celestial pole is always above the horizon, as are many of the constellations around it. From my location that includes Cassiopeia, Ursa minor, most of Ursa major.
More generally, for Northern Hemisphere observers, the further north a point is, the longer it will remain visible.