The private pilot was conducting touch-and-go takeoffs and landings. He reported that during his second landing and subsequent takeoff, he felt a vibration in the nose landing gear and decided that he would make the next landing a full stop and taxi back to the fixed base operator. During the third landing attempt the airplane touched down, the pilot heard a “snap,” as the nose landing gear settled onto the runway, and the nose landing gear then collapsed. Security video of the landing showed that the airplane landed normally, and after several seconds of landing roll, the nose landing gear collapsed and the airplane came to a stop on the runway. The airplane’s firewall, engine mounts, and lower fuselage structure were substantially damaged. The airplane was subsequently removed from the runway and stored in a hangar at the airport. The hangar later caught fire and the airplane was destroyed. An examination of the nose landing gear components could not accomplished.