Say we have a spherical polygon consisting of a series of vertices $v_1, v_2, \ldots, v_i$ on the unit sphere connected by great-circle arcs and are given a point $X$ on the unit sphere which is guaranteed to be inside the polygon (thus defining an interior). In general, the polygon may not be convex. Computationally, we could pick points and use an intersection-counting algorithm to determine whether they are inside of the polygon. But I'm wondering whether there is a point which could be directly calculated from the information given which would not be inside the polygon.
I haven't thought about this part as much but bonus points if you can find a point that is in some sense "far away" from the polygon.