C172 pilot arriving GAI reports NMAC with another C172 departing Runway 32R. The pilots were in radio communication with each other but the departing aircraft turned an early crosswind causing the conflict.

Narrative

While approaching and having announced '3 mile 45 to right downwind GAI;' I heard the other aircraft; familiar to me; announce; 'Taking off; Runway 32; Montgomery County.' Piloting the plane on the runway; I returned my attention to setting up downwind entry. Just upon entering downwind; I spotted Aircraft Y climbing from below and to my right ahead. Slowing and watching Aircraft Y; I saw it pass us above and continue outbound; apparently on a departure course to nearby EMI VOR; from a crosswind off a very short upwind leg. As I observed Aircraft Y; continuing and announcing myself on downwind I noted that Aircraft Y turned to a course parallel to mine; ahead of me and at a higher speed. Announced; 'Airplane on downwind; do you see me? To which I answered; 'Affirmative; you are ahead; off my left wing.' Then; just as I was abeam the numbers of Runway 32; Aircraft Y announced; 'Airplane on downwind expedite your turn to final;' and begin to and announced; 'We are turning base.' Realizing the further conflict could arise as Aircraft Y cut across my path; I announced; 'Aircraft X is breaking off; I will follow you in.' Aircraft Y subsequently continued the landing as a 'touch-and-go.' I continued on a slightly wider than normal and extended downwind for safe separation to my planned; normal full stop landing.