let $x_1$, $y_1$, $x_2$, $y_2$ be points on the sphere $S^n$ such that $x_1$ and $y_1$ are not antipodal and similarly for $x_2$, $y_2$. Why $x_1$ and $y_1$ can be joined by a path that do not cross with a path joining $x_2$ to $y_2$? and why we call these paths "shortest geodesics"?
If two points on a sphere are not identical and not antipodal then they lie on a unique great circle. The shortest distance between the two points will the shorter of the two arcs of the great circle between the two points
Two distinct great circles intersect twice. The two shorter arcs between the pair of points may or may not intersect. This diagram may help.