In order to measure the polarization of incoming gravitational waves and avoid having an orientation to which the telescope is insensitive, a minimum of two detectors are required. While this could be done with two 90° interferometers at 45° to each other, the triangular form allows the arms to be shared. The 60° arm angle reduces each interferometer's sensitivity, but that is made up for by the third detector, and the additional redundancy provides a useful cross-check.
