BiTeivesAsoroh
On the eighth of Teves 3515, the Torah was translated (written) to Greek in the days of Talmai the King. The world was dark for three days. On the ninth day of Teves it is not known which calamity occurred: Shulchon Aruch 580:2, In our Selichos it says that Ezra HaSofer was Nifter (died) on the ninth of Teves. - Mogein Avrohom #6
One is obligated to fast etc. on the tenth of Teves because of the bad that took place on them. This is to awaken the hearts to Teshuva - (Repentance): (Rambam) Mogein Avrohom
With such a bundle of calamities, what must be one’s outlook for Teshuva? We may be able to say, that since we are rectifying our actions at this time, it should be actions of correction and rebuilding. In contrast to the decline that took place in history in this time period.
I shall remember the pain that has befallen me, in this month three calamities have befallen me, etc.... The Selichos, said on the fast of the Tenth of Teves, reflects in its start the tragedies that we suffer from until this very day. The paragraph of this portion of Selichos ends, He says to the ocean till here you shall go. Even if it is a month of misfortunes, it is until here. The forthcoming months are ones of Hischadshus and Geuloh (renewal and redemption).
The Rebbe Reb Bunim ZY”A of Parshischoh said, Hayom! (today), Im BiKolo Tishmoh’u (if we will listen to His voice). We will merit the ultimate redemption, with the coming of Moshiach.
On the tenth day of Teves, the king of Bohvel laid siege on the Holy City of Yerushalayim. The Chasam Sofer says that since Asara BiTeves is the day that the destruction of the Beis HaMikdosh started, it is also a day of judgement if we are going to merit its rebuilding this year.
And so, we conclude the Selichos...
He redeems and comforts, saves from grief, acts for the sake of His people, my soul, the fast of the fifth, for joy and happiness, the fast of the fifth, for relief, for sanctity, the fast of the tenth and the fourth – Selichos for Asarah B’Teves