We are asked to find which option is false in spherical geometry. Let's analyze each option separately.
Option A
Option A
The shortest path between two points on a circle is an arc.
The shortest path between any two points is an arc of a great circle through the points. This statement is independent of whether or not these points lie on a circle. Therefore, option A is true.
Option B
Option B
If three points are collinear, any of the three points lie between the other two.
Remember that a line in spherical geometry is a great circle. Let's analyze three points which are collinear — that lie on one great circle.
Note that we have two arcs between point B and C. One arc contains Point A, the other one does not. This same is for any two other points. Therefore, option B is true.
Option C
Option C
A great circle is infinite and never returns to its original starting point.
A great circle is obviously a circle. Therefore, it cannot be infinite. Is always returns to its original starting point. This tells us that option C is false.
Option D
Option D
Perpendicular great circles intersect at two points.
Any two great circles intersect at two points. This tells us that also this is the case for perpendicular great circles. Therefore, Option D is true.
Conclusion
Only Option C is false. Therefore, the answer is C.
