The student pilot reported that following an uneventful flight, as he entered the airport traffic pattern on the downwind leg, he was notified by another pilot that there was an airplane operating within the traffic pattern with no radio. The student pilot established visual contact with one airplane; however, he did not establish visual contact with the airplane operating without a radio until reaching the mid-field point on downwind. The student pilot stated that throughout the approach to landing, he maintained visual contact with the airplane until it would not be a factor. The student pilot subsequently landed the airplane with the landing gear in the retracted position, which resulted in structural damage to the fuselage. The student pilot stated that his attention was diverted between maintaining airspeed and a stable approach to the runway, while maintaining visual contact with the airplane that was not in radio contact. He added that he failed to do his before landing checklist as he normally does at the mid-field point on downwind and each turn in the traffic pattern. The pilot reported no preaccident mechanical malfunctions or failures with the airplane that would have precluded normal operation.