Understanding the "Break" in Birkat Hamazon: A Halakhic, Aggadic, and Historical Journey

Why do we answer אמן to our own recitation of the third berakhah of Birkat Hamazon? The answer is halakhic and has to do with the structuring of berakhot in series. Ashkenazim and Sephardim have slightly different approaches. But according to all, Birkat Hamazon doesn't really follow the rules. To understand what's going on behind the halakhah, we need to understand how the Sages imagined the history of Birkat Hamazon and how they viewed the fall of Betar in the year 135 - an event that we commemorate on Tisha B'Av as well as on Tu B'Av.