How are profits distributed among three brothers after the third brother has managed the shop, and how is the value of the third brother's purchase of the shop calculated to relieve the other two brothers?

The contract between the two partner brothers and their third brother can be either an ijarah (leasing/hiring) or a mudarabah (profit-sharing partnership).

If the contract is ijarah: The working brother is entitled to his agreed-upon wage, and the shop's profit is divided between the two owning brothers according to their agreement. The working brother has no right to this profit because he is an employee (ajir).

If the contract is mudarabah (partnership): The profit will be divided among the three brothers according to what they stipulated in the contract.

If the third brother wishes to purchase the shop to own it, there is no harm in that. He should pay his two brothers the agreed-upon price for their share, regardless of whether he was a partner with them or an employee.