Nebuchadnezzar II of Babylonia sieged Jerusalem on the Tenth of Tevet. Eighteen months later, on the 17th of Tammuz, he broke through the city walls. Three weeks later, on Tisha b’Av, the siege ended with the destruction of the Bais HaMikdash and the conquest of the Kingdom of Judah. The Tenth of Tevet is thus considered part of the cycle of fasts including Shiva Asar b’Tammuz (17 of Tammuz) and Tisha b’Av (9 of Av).
On Chanukah, we light one flame then two then three… until we have eight flames. Night after night, we, so to speak, increase the miracle. Or, more accurately, we recognize the increasing magnitude of a miracle. Similarly, we do the same on the Tenth of Tevet through Shiva Asar b’Tammuz (17 of Tammuz) and Tisha b’Av (9 of Av). In theory, we could merely recognize the destruction of the Bais HaMikdash, Tisha B’Av. But we do not. We recognize the entire process of this destruction including the three milestones of this destruction: Asara b’Tevet, Shiva Asar b’Tammuz, and Tisha B’Av. (And the Fast of Gedalia but, in the name of simplicity, we’re not including this Fast.) We recognize that the destruction of the Bais HaMikdash was not a single, superficial event. This destruction started relatively small and increased in magnitude until there was utter and complete destruction. These three fasts, if looked at properly, force all Jews to recognize the depth, breadth, and graduality of the destruction of the Bais haMikdash.
According to the Rambam, the focus of these fast days, besides merely recognizing the historical significance of these events, is teshuva.
“The essential significance of the fast of the Tenth of Tevet, as well as that of the other fast days, is not primarily the grief and mourning which they evoke. Their aim is rather to awaken our hearts towards repentance; to recall to us, both the evil deeds of our fathers, and our own evil deeds, which caused anguish to befall both them and us and thereby to cause us to return towards the good.”
On this day and the days that follow, we need to do teshuva.