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?
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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.
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