Your situation is a difficult one, and presents several different sets of questions. Some of those questions concern coverage for your brother's upcoming nursing home care. Other questions concern whether
and how you and your sister can get repaid the money you've lent your brother. Let's take them one at a time.
First, there's the question of who's going to pay for your brother's nursing home care. Medicaid considers a person's income and assets when determining whether it will cover nursing home costs. In your brother's case, whether he qualifies for Medicaid nursing home coverage, and when, will depend in part on how much money he gets from the sale of the family house. As long as he owns a one-third interest in the house, until it's sold, he probably won't even qualify for Medicaid nursing home coverage. Once the house is sold, he will have to spend almost all of the proceeds he gets from the sale of the house -- and the spending must be either payment to the nursing home or other legitimate expenses -- before Medicaid will begin to cover him. Once he's spent his assets, Medicaid will then consider covering his nursing home care if his income is not too high. (If he repays you and your sister from the proceeds of the house sale, he'll qualify for Medicaid that much quicker.) If he continues to receive income, that will also have to be paid to the nursing home, except for a small allowance for personal expenses.
Until your brother qualifies for Medicaid and Medicaid begins actually paying for his care, Medicaid will not have any claim on your brother's portion of the proceeds from the sale of the house. So, if you sell the house before he becomes eligible for Medicaid, there is no reason why he can't repay the debt he owes you and your sister out of his portion of the house sale. Because of his Alzheimer's, however, there's a question about whether he is, or will be at the time, competent to acknowledge the debt and pay you (or even execute the sale of the house). If he's still competent to make such decisions, you and your sister may want to help him execute a Power of Attorney for Finances as soon as possible. This will allow you, or whomever else your brother appoints, to execute the necessary documents for selling the house and paying you and your sister what he owes you, as well as providing for someone to legally oversee his continuing finances once he's no longer competent to do so himself.
If your brother is already at the point where he's no longer competent to make decisions about his finances, you may need to get help from a court in order to handle these matters, through a guardianship or conservatorship. Your brother's condition may make it difficult for you and your sister to accurately determine your brother's legal competence, and therefore to decide the right course of action. So, it's probably a good idea for you to seek the advice of a lawyer who specializes in elder law, family law, or probate law. This is particularly true because the decisions you and your sister, and your brother if he's still able to participate, may have long-term consequences during the course of your brother's illness.