The pilot completed the before landing check on left base leg, by lowering the landing gear, and extending approach flaps. He did not verify the landing gear handle was in the full down position. The pilot turned onto final approach, lowered the flaps to the full down position, and did not verify the landing gear handle was in the full down position. The nose gear collapsed on landing roll out followed by the main landing gear. The pilot raised the flaps as the airplane started to skid down the runway. The airplane came to a complete stop, and the pilot observed the landing gear handle was not in the full down position. Examination of the airplane by maintenance personnel revealed no mechanical defects with the landing gear or the landing gear selector switch.