Bonanza pilot reported an NMAC while entering the traffic pattern after another aircraft climbed from directly under the reporter's aircraft requiring an extended downwind to avoid a collision.

Narrative

I departed ZZZ1 at approximately XA:29 in a Beechcraft Bonanza. My passenger and I were flying to ZZZ to eat at a restaurant on the field. The flight was very smooth and we experienced no problems while on route. My passenger was new to small airplanes and I spent quite a bit of time explaining various instruments and procedures. We also flew through the edge of a high intensity training area so we spent extra time watching out for traffic. Approximately 50 miles out from our destination; I received the ATIS and tuned to ZZZ UNICOM/CTAF. We then began a slow and steady descent. We observed what I took to be a female student doing touch and goes at ZZZ both on the radio and on my tablet via Foreflight traffic. I made my initial call around 12 miles out to notify any other local traffic of our intentions to land. I made a second call around 5 miles out at which time the student had left the area. The only other traffic in the vicinity was a King Air that was a few miles behind us also inbound for ZZZ. They stated that they would keep an eye out for us and follow us in once they made visual contact. I then announced that we were entering a right traffic pattern on a textbook 45-degree angle on the downwind side. While setting up for the traffic pattern; I noticed on my tablet what appeared to be another airplane virtually right underneath us with a -100 next to it. At first; I thought that the ADS-B had picked up our shadow on the ground and it was a false positive traffic alert. As I studied it closer; I noticed that its previous track was slightly different than ours and that it had a more straight-in; southerly track rather than a 45-degree pattern entry like ours. Aside from the student we had heard earlier and the King Air behind us; there had been no other traffic reported in the area and certainly no one announcing they were entering the traffic pattern. At the exact moment I realized there might be a problem; a voice came over the radio announcing that an RV was at midfield at ZZZ intending to land on XX. There indeed was another airplane directly below us that had made no radio announcements until then. I then had to decide whether to immediately change course to the left and risk a head on collision with a King Air that we could not see; or find a safe alternative way to distance ourselves from the unseen and previously unannounced aircraft. I began drifting a bit to the left to put some distance between us and where the aircraft likely was and at the same time announced over the radio to ask if there was an RV in the vicinity and where they were located. At that exact moment; to my astonishment; a small; yellow RV airplane with a sliding canopy appeared to pop up (gain altitude) just off of my right wing parallel with us. He announced over the radio that he was in the pattern at midfield and would do a short final. I then announced that I would execute an extended downwind leg to give him ample time to land safely. I never once saw him look back and look for us even after telling him we were right over him (I do not think he ever saw us). The King Air behind us said that they too would extend if needed and follow us in. The RV then landed with us following shortly after him. We rolled farther than him and thus parked at the terminal first. The entire time I kept wondering if I had imagined him gaining altitude or if perhaps it was I who had unknowingly descended. For this reason; I did not confront the other pilot without hard evidence. I did note his N number with the thought that I would check ADS-B data when I returned home and had a chance to look it over. While there is no way to prove radio procedures that were or were not followed; sure enough the data shows us arriving at the same time and his airplane with a sharp increase in altitude (the pop-up I thought I witnessed) as he approached midfield. What I still cannot believe is that despite not making any radio announcements until at midfield (versus the 4 thatI made); even after I announced that we were at the same position at the field and even with a glass canopy that gave him 360 degree viewing range; he somehow decided to pull up! All he had to do was look up and to the left and he would have seen us. And why on earth would you change your altitude immediately after another plane radios in that they are at the exact same position. I firmly believe that had I not drifted to the left; this gentleman would have killed us all with his little pop-up stunt. I do not want to overgeneralize; but in my previous 32 years of flying as a passenger and 13 as a student/private pilot; the only times I have ever felt unsafe are always because of the actions of older pilots; typically male; in RV or homebuilt type aircraft that don't make radio calls; don't follow procedures; and just seem to do whatever they want with no regard to anyone else's safety. They act as if rules don't apply to them and they can do whatever they want. ZZZ is a fairly busy airport; especially at meal times due to the field's restaurant. To not make any calls with the amount of traffic that airport receives is extraordinarily dangerous. Safety is my top priority at all times. I never skip a preflight and use checklists extensively. I announce my intentions by the book and make all my calls on every leg regardless of how busy the area I'm flying in is. I understand that it can sometimes be overkill and you don't want to clog up the frequency; but to not talk at all is lunacy and is a death-wish at a busy field. Until the RV pilot gets back up to speed on simple radio procedure and traffic pattern maneuvers; he puts himself and others at risk.