Taanis, Chapter 2, Mishnah 10. The Mishnah opens with a ruling: "Ein gozrin taanis al hatzibur b'Rosh Chodesh, b'Chanukah u'v'Purim" - on these days a public fast is not decreed to begin with.
A series of fasts already underway:
If a series of fasts has already begun, and one of them falls on one of these days - Rosh Chodesh, Chanukah or Purim - "ein mafsikin" - the series is not interrupted, and the community fasts even on these days. This is the opinion of Rabban Gamliel.
Rabbi Meir's qualification:
Rabbi Meir adds a qualification: even though Rabban Gamliel said that the series of fasts is not interrupted when it falls on Rosh Chodesh, Chanukah or Purim, "modeh hayah she'ein mashlimin" - one does not fast the entire day until its end, but rather breaks the fast partway through.
Tishah B'Av that falls on erev Shabbos:
The same applies to Tishah B'Av that falls on Friday. With our fixed calendar this cannot happen, but in the days when the month was sanctified by the testimony of witnesses and there was no fixed calendar, Tishah B'Av could indeed fall on erev Shabbos. Here too, in Rabbi Meir's opinion the fast is not completed; rather, it is broken as Shabbos approaches, so that one does not enter Shabbos in a state of hunger.
The halachah: We do not rule like Rabbi Meir. Rather, whenever the obligation to fast applies, one fasts the entire day until its end. And the same applies to Tishah B'Av that falls on erev Shabbos.