According to the pilot, he flew the traffic pattern to a touchdown zone landing on runway centerline. However, after touching down, the airplane "floated back up," then touched down again "with authority." It bounced again, then "weather vaned some," and the pilot became concerned that the airplane was "chewing up runway." The airplane then touched down a third time, but the pilot did not pull back on the yoke. He subsequently heard the propeller strike the runway, the airplane then veered off the right side of the runway, the nose landing gear collapsed, and the engine mounts broke. The pilot indicated that there were no preexisting mechanical anomalies that would have precluded normal operation. Nearest recorded weather included winds that were about 90 degrees from the right of the runway, at 10, gusting to 15 knots.