How to buy brother out of our house?

Based on your email, the two of you never put together any agreement when you bought the house addressing how any future buyout would occur, and the terms of that buyout, including potential increases in value, who paid for the improvements, and who paid the mortgage. That means you might be in a mess right now, depending on what the other brother thinks. Ultimately, the value of his portion of any downpayment or contribution has value, even at a very low interest rate, over the past 8 years. That is countered by your contributions in maintaining the house. Plus, with the market over the past year or so, there may be some increase in the value of the house generally. What you need to do is hire an attorney to assist you to fully spell out the terms of the sale to you, particularly as contracts between family members are often looked at by the courts as suspicious, and you'd have a very large burden of proof is your brother challenged the contract at a later date. So, avoid futher headaches and hire an attorney on this. I could assist you, if you desire.