I am trying to take some state and county data that I downloaded from various gov't sites and reproject it to our local coordinate system. The basedata is mostly in Lambert Conformal conic, with either GCS_North American_1983 or GCS_North_American_1927 Datum. When I try to reproject the basedata layer, it says that it was projected, but the layer no longer draws in ArcMap. I can't even zoom to the layer. The local grid has a false easting of 43517.320127 ft, a false northing of 41291.365728, is Transverse Mercator projection, and WGS_84 Datum. Even if I add the unprojected basedata to a map in the local coordinate system and use a transformation to to let ArcMap project on the fly, the layer does not draw. What am I doing wrong?