We have used light echoes from the inner few arcseconds of the region around SN 1987A to define the three-dimensional geometry of its circumstellar nebula. SN 1987A appears to be surrounded by a double-lobed nebula, the waist of which is nearly coincident with the prominent elliptical ring, 1.''7 in major axis length, seen prominently in atomic recombination lines. The symmetry axis of this double-lobed structure corresponds closely to the axis which would also imply that the ring is actually circular and just seen as an ellipse in projection. By confining the range of ring geometries, this supports previously described techniques using the Panagia et al. light travel time measurement across the ring compared to its angular size to produce an estimate the distance to SN 1987A. With corrections for the distance between the SN and the LMC disk at the end of the galaxy, as well as the correction to the center of the LMC caused by the tilt of the galaxy, one can infer a distance to the LMC of 51.9 +/- 3.1 kpc (or 52.9 +/- 2.6 kpc if one simply assumes that the ring is circular). We also note that the red supergiant wind contains a density concentration in the ring plane, as anticipated by hydrodynamic models for the double-lobed nebula. We put observational limits on the presence of a companion star which might be responsible for these anisotropies. We also discuss implications of HST WFPC2 observations of recombination radiation from this region.