Stephen:
Yes, you do need a .prj file if you expect ArcGIS Explorer (or ArcMap) to reproject your data on the fly. If you are using ArcMap, you can find prj files here: C:\Program Files\ArcGIS\Desktop10.0\Coordinate Systems\ Alternatively, if you do not have a prj file with your shapefile "triplet", you could change the map projection in AGX to match the projection of the shapefile.
See attached zip file for the projection file you are looking for.