Puzzle of the Week #42 - TriBall

TriBall is a sport which is played between two teams of three players.

My local TriBall club has an annual tournament.

In the tournament, every possible trio plays every other possible trio exactly once each. Luckily the ‘matches’ are only a couple of minutes in length, but even so the tournament takes a while to complete.

The club has one more member this year than it did last year. As a result, the total number of matches in the tournament is exactly three times as many as last year.

How many members are there in my TriBall club?