It should not be too difficult to implement this buyout of a partial interest in your inherited house, but the laws of each state are different. For this reason I
would at least consult with an attorney for an hour or so, just to be sure that you are doing it right. You don’t want to mishandle the situation and face a much stickier legal problem years down the road.
In part the process differs depending on whether the property is still in the trust, or if it’s been distributed out to the three of you as individuals. If it is has been done as individuals then generally the way the transfer is done is via a grant deed, signed by the selling brother and transferring his interest to the buying brother. You can also have a purchase and sale agreement documenting the amount of the payment, but that isn’t necessary. Then, the deed will need to be recorded in the county recorder’s office of office of deeds for the county in which the property is located.
If the property is still held in the trust, then typically you would first have to distribute the property in equal one-third shares to the three of you, by a deed signed by the trustee. Then, after the property has been deeded out of the trust you can follow the ordinary deeding procedure laid out above.
The general tax rule is that the value of the property at the date of your dad’s death comes to each of you tax free. There should have been an estate tax return filed that sets that amount, and your brother would only owe capital gains tax (not income tax) on any increase in value above that amount. Your questions states that it has lost value, so this won’t be a problem for you!
There’s another aspect of the situation that you haven’t asked about, which is equally important, and that is what are your co-ownership arrangements going forward. You and your brother will now co-own the property, in unequal shares, and you should put together a simple co-ownership agreement for the future. That agreement would describe how income and expenses are going to be shared, and how decisions are going to be made. This is an agreement you can probably write up yourself, but it wouldn’t hurt to have it reviewed by a real estate lawyer who has experience in co-ownership agreements.