How to prove geometrically that the altitude of Polaris is equal to the latitude?
I have seen in handouts that the altitude of the pole star gives the latitude of that place.
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This is a consequence of geometry: consider a cross section of the Earth through the poles, so that it can represented as a circle. Given a place at latitude $\lambda$, the horizon is equal to the tangent of the circle at that point. The tangent of a circle makes a 90 degree angle with the 'line from the place through the Earth's center', so the angle between Polaris and the horizon (Polaris' altitude) is the same angle as between the equator and the 'line from the place through the Earth's center'.
I've drawn Polaris quite close to Earth; the reason this works for all places is that in fact it's very far away (compared to the Earth's size), but always in the same direction (straight up North).