The pilot reported that during entry to the circuit at Rottnest he initially joined for a landing on runway 27 and he had lowered the landing gear early to help slow the aircraft down. During the circuit join the wind changed and he then manoeuvred for a landing on runway 09. He had to fit in with two other aircraft also manoeuvring for a landing on runway 09. The pilot made a high approach for runway 09. Prior to the approach he had completed his downwind checks which included placing the landing gear lever in the down position and obtaining green indications. During the flare the landing gear warning horn sounded and he applied full power but it was too late to avoid a touchdown. The pilot reported that a witnesses on the ground stated that the landing gear appeared to be in the down position but that it collapsed on landing. An inspection of the aircraft at Rottnest Island indicated that there was no scuffing on the main wheel tyres as there would have been if the gear had rotated, on the ground, prior to retracting. The nosewheel doors were slightly open and the damage to them indicated that they were opening rather than closing. There were a series of propeller strike marks for a distance of 350 ft along the runway. The depth of the marks indicated that the engine had been at a high power setting and that the aircraft had flown parallel with the runway for most of the 350 ft. An inspection of the landing gear, after recovery, did not disclose any evidence of collapse. The evidence indicates that the gear was probably up when the warning horn sounded during the flare and it was in the process of extending when the aircraft touched down.