Are you allowed to take a shower on a fast day, and is it suggested? Once we are talking about bathing, what about going to the Mikvah on a fast day? Does it make a difference if one goes daily, and/or one is a Mohel on that particular fast day and the custom is to go to Mikvah the day you preform a Milah?
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The Gemara in Megilla 6a says Rebbi Yehuda Hanasi went to the bath house on 17 Tammuz.
The Shulchan Aruch OC 550:2 rules that bathing is permitted on the 'minor' fasts and is only forbidden on Tisha B'av and Yom Kippur.
The Mishna Brura there (sk 6) says that a meticulous person ("baal nefesh") should be stringent on all the 5 afflictions of tisha b'av (eating, washing, anointing, wearing leather shoes and conducting marital relations) even on the minor fasts with certain exceptions for mitzva purposes such as bathing in warm water for Shabbat when 10 Tevet falls on Friday.
In the Shaar HaTziyun there (sk 8) he quotes the Ateret Zekeinim who says there is a custom not to bathe in warm water on the minor fasts.
So, it seems a mikva is clearly permitted as it is for mitzva purposes and it is not usually warm (it is certainly not bathing for pleasure.)
In terms of a regular shower, it is certainly permissible according to the basic law. Some may take it upon themselves to be strict regarding warm water.