Dude... forget drawing lines, imagine a plane that passes from the 2 points and the center of the earth. There is only one such plane (3 points define a plane). The section of that plane with the surface of the earth is the 'great circle' between those two points.
Check google earth (not google maps) and click 'measure'. The distances and routes it gives you are great circles. Even at same latitude it doesn't follow a line of latitude (unless both points are on equator)
Alternatively:
Take an orange. Mark 2 points. Then imagine cutting with a knife that passes from the two points and the centre of the orange. The cut is a 'great circle'.
Geometry...