PA-28 engine pilot reported an NMAC while entering the pattern at a non towered airport.

Narrative

My flight originated at ZZZ. I had just completed the ZZZ route into ZZZ1 and was headed northwest towards ZZZ2; for fueling. The weather was clear with excellent visibility >10 miles. Approximately 10 miles south of the field I announced my location and intent to land at ZZZ2 using Runway XX. At that time; I overheard two planes in the pattern discussing changing the active from Runway XY to XX due to a change in light winds. Approximately five miles south of the field I announced my location; indicating I would overfly the airport midfield at 2400 feet; (1000 feet higher than the pattern altitude of 1400 feet); and then enter a 45 for a right downwind to Runway XX. Prior to reaching ZZZ2 I heard another aircraft announce he was east of the field and would be entering a 45 for a right downwind for Runway XX. I crossed over the field at 2400 ft and flew northwest of Runway XX approximately 1.5 miles and started my counterclockwise descending left turn with the intent to enter the right pattern midfield on a 45 to the downwind of Runway XX. I was descending through approximately 1800 ft and about to announce my location when my TCAS suddenly activated indicating 'Traffic; 12:00; low.' I immediately turned to the right and then noticed another aircraft; approximately 150 ft horizontal and 150 feet below my path. This aircraft was on the right downwind to Runway XX. I continued in a clockwise turn to gain spacing and then had a second Traffic alert for a second aircraft also on the downwind for Runway XX. I followed that aircraft on the downwind and landed without incident. Luckily; I had the TCAS alert; otherwise I suspect I could have had a mid-air collision. Reviewing everything that happened it appears I committed the following errors. I incorrectly believed the aircraft that announced he was entering the 45 for the downwind was actually on the 45 and I had proper spacing. I reviewed the ADS-B tracking and noticed aircraft at ZZZ2 fly a much wider pattern than I am used to flying; possibly due to an 800 ft hill on the downwind/final leg for Runway XX. I should have greatly extended my northwest position from the airport prior to attempting my descending left turn to enter the pattern on the 45. I failed to perceive I could have been so close to the pattern traffic on my descending turn. This was the first time I have ever landed at this airport; and I was solely responsible for the near miss incident. Although most of my flying experience is in helicopters; I still have about 440 hours in fixed wing aircraft; mostly local. I have definitely learned from my mistakes in this incident.