From the original post
So between 2 points where a great circle ain't formed,we have a small circle.
There are no two points on the earth which cannot be joined by a great circle.
Now the smaller arc of this small circle would be the shortest distance between these 2 points.
Try testing this argument on a football or other suitably small sphere.
Draw a circle with a diameter that is less than that of the sphere. Draw dots at two points on this circle.
Draw a straight line across the circle joining the two points.
You now have three paths between the two points. One is the straight line and the other two are arcs of the circle.
If you measure these paths you will find that the shortest is the straight line.
Now extend the straight line until the two ends meet. You will find that you have drawn a circle that has its centre at the centre of the sphere. This is the great circle on which the two points lie.
So the shortest distance between the two points is part of the great circle on which the two point lie.