Consider a line through a point p on a sphere, now take another line passing through that point. It is found that the second line always intersects the first line in antipodal points. How do I write a mathematical proof that this is always the case?
I'm trying to prove it myself but I can't figure out how to begin.
A "line" on the sphere (also known as a "geodesic" or a "great circle") is the intersection of the sphere with a plane through its center. Thus the two great circles have two such planes, both of which pass through the sphere's center, and therefore intersect. That intersection is a (Euclidean) line through the center of the sphere, and it passes through the sphere at the intersections of the two great circles.