During the descent to the destination airport, the pilot noticed that another airplane had departed from the runway and was climbing to enter the traffic pattern. The pilot decided to extend his traffic pattern entry to visually observe the other airplane and to provide separation so he could land. During this time, the main landing gear warning horn activated due to the engine power reduction that occurred, and the pilot then activated the main landing gear warning horn mute switch. The other airplane performed a touch and go, the pilot forgot to extend the main landing gear, and the pilot proceeded to land the airplane. During the gear up landing to the dry asphalt runway, the airplane departed the runway to the right, impacted two runway signs, and came to rest upright on a flat grass field. The airplane sustained substantial damage to the underside of the fuselage. The pilot reported there were no preimpact mechanical malfunctions or failures with the airframe or the engine that would have precluded normal operation. The pilot further reported that he should have performed a final landing check on short final and that he allowed the avoidance of the other airplane to distract him from his normal landing procedures and flow.