The "triangle" thing is a good argument for proving the earth is round, there is a flaw in it though.  WHen you see it demonstrated by someone holding a styrofoam ball or a basketball or some other sphere...they usually 1) start at the top of the sphere, 2) go down to the equitorial line of the sphere (a distance of 1/4 circumference), and then 3) turn 90 degrees to move along the equatorial line a distance exactly 1/4 circumference, and then 4) turn 90degrees to end up back at the starting point at the north pole.   This does indeed work on a sphere but only if you use the precise distance of 1/4 circumference for each line.  use more or less than this and you will not end up back at the beginning.  This is why if you go in your back yard and walk 10 ft., turn 90 degrees and walk 10 feet, then turn another 90 degrees and walk ten feet you will not end up at the starting point...because you did not use a distance each time of 1/4 the supposed circumference of the earth.

In other words, the only way to prove/disprove this theory on earth is to carry out the experiment by traveling 1/4 the circumference of the earth at each 90 degree turn.  No one has ever made this journey.  Until that happens, the triangle explanation is unproven.