Generalization of tangents to circle theorem.

Question

There is this commonly known and easy to prove theorem that if you have a circle $O$ and two lines that are tangent to $O$ (one of them is tangent at some point $A$ and the other one is tangent at some $B$) and they meet at point $P$ outside of the circle then $|PA| = |PB|$. I was wondering if there is some generalization (or similar theorem) of this theorem to $3$ dimensions, so a sphere and three planes tangent to it and having something in common. Have you heard about anything like that?

Answer 1

For any $2$ points on the sphere $S^2$, there exists a greater circle that goes through both of them. If two of the tangent lines intersect, they lie on the defining plane for this greater circle, and hence the $2$ dimensional version of the theorem applies.