Two great circles intersect at two antipodal points (Spherical Geometry)

Question

I've been looking into spherical geometry and have seen that two great circles intersect at exactly two antipodal points. Visually, I understand why. However, I have yet to see a rigorous proof of this; is there a rigorous proof of this?

Answer 1

A great circle by definition is the intersection of the sphere with a plane passing through the center of the sphere.

Let $C_1$ and $C_2$ be two distinct great circles. Let $\pi_1$ and $\pi_2$ be two planes passing through the center of the sphere such that $\pi_1\cap S^2=C_1$ and $\pi_2\cap S^2=C_2$.

Thus $C_1\cap C_2 = S_2\cap (\pi_1\cap \pi_2)$.

Note that $\pi_1$ and $\pi_2$ are distinct planes and hence they intersect in a line. This line passess through the center of the sphere since both the planes contain the center.

Therefore $C_1\cap C_2$ is $S_2\cap \ell$ where $\ell$ is a line passing through the center and hence $C_1\cap C_2$ is a pair of antipodal points.