Why do detectors for gravitational waves have only two perpendicular arms, not three?

LIGO's arms are 4km long, which makes the problem even worse. Besides, you can just move a quarter of the way around the planet and build another 2-arm facility there, and it'll automatically be at right angles to your first facility (and as long as you do know their relative positions and orientations precisely, you don't have to put them at exactly right angles to each other).

The basic idea is that you have destructive interference. This is hard(-er) to achieve with three photons as well as I don't see a trivial way to do the beam-splitting.