Blue line, duuhhhh.
It's the same reason why the equater goes around the earth, and not tangentially off into space.
but. yeah.... This is a simply one. Suppose two people hold a rope and stretch it until it has zero sag. The rope seeks to find the shortest distance between the peeps. That distance is a straight line. Now, the fellas start to back away from each other until until they pass beneath the horizon. At that point, the rope is touching the earth at a spot equidistant from our two friends. As these guys continue to back away from eachother, the rope continues to touch the earth at more and more points.
Then again... Once they disappear over the horizon, won't the rope begin to slip off to the side unless a third friend is holding it at the midpoint? Remember, the tension in the rope is attempting to relieve itself by finding the shortest distance between the two rope jockeys?
And a final question... Once the rope circles the earth, can you jump the rope? Or is the rope jumping you? This part has me LOLing at myself. Once you get the rope turning, would it remain in motion forever in orbit?