I am trying to learn spherical geometry, but I have difficulty resolving a simple issue.
Let's define a sphere's equator and it's poles N, S. if we create a great circle by tilting the equator circle by degree of $\alpha$. In that great circle point P is the closest to N (the point in the sphere where the latitude is highest). Now, let's look at the great circle that is created by the points N, P on the sphere. Those great circles intersect in point P on the sphere.
My question is about the angle between those great circles at the point P. It seems to me that the angle is indeed $90$. but according the definition of this source (point 9):
"By the angle between two great circles is meant the angle of inclination of the planes of the circles."
It seems that the planes of the great circles are not perpendicular - meaning the angle between them is not 90.
I would appreciate if someone would instruct me what I'm missing here, and how to show, formally, that the angle is 90?