On July 18, 2005, at 1300 eastern daylight time, a Piper PA-34-200, N143PF, registered to KD Aviation Enterprise Inc., operated by Falcon Aviation as a 14 CFR Part 91 instructional flight, went off the left side of the runway at Peachtree City/Falcon Field on Landing roll out. Visual meteorological conditions prevailed and an instrument flight rules flight plan was filed. The airplane received substantial damage. The commercial pilot rated flight instructor (CFI) and private pilot receiving instruction reported no injuries. The flight originated from Athens, Georgia, on July 18, 2005 at 1205. The CFI stated the private pilot was flying the airplane from the left seat and had just completed a simulated single engine approach. The airplane touched down on runway 31 and the CFI informed the pilot to apply brakes so he could turn off at the mid filed taxi way. The pilot applied brakes and the airplane veered to the left. The CFI verbally instructed the pilot to get off the brakes and completed a positive transfer of the flight controls while applying right rudder. The airplane continued to veer to the left. The CFI applied full right rudder, and assumed the airplane kept going to the left because the left main landing gear tire was flat. The airplane went off the left side of the runway and the left main landing gear collided with uneven terrain and collapsed. Examination of the runway revealed skid marks were present 1,400 feet from the approach end of the runway and extended 345 yards to where the right main landing gear left a gouge in the sod area west of the runway. Examination of the left main landing gear tire revealed the tire was flat with a flat spot and hole present in the tire. A functional check of the brake system revealed no anomalies.