If the two lines are coplanar, then it is a third, parallel, line which is halfway between the two lines.
If the two lines are not in the same plane, then I think it will be a saddle shape (paraboloid).
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They are parallel lines
Yes. All latitudes are approximately 111 km apart.
They are parallel lines
The perpendicular bisector of the line joining the two points.
Parallel lines could fit the given criterior