Is it considered usury for a person to invest their money in a friend's business, with an agreement to share in profits and losses, without setting a fixed percentage for the profits, and to receive a check for the original amount as collateral?

Paying money to a person for investment in exchange for a common share of the profit is considered Mudarabah (definition) (profit-sharing partnership), which is one of the permissible types of partnerships. Ibn Qudamah mentioned that there are four permissible types of partnerships, among them Mudarabah. Al-Khiraqi stated that the profit is distributed according to what the two partners agree upon. If your friend does not guarantee the capital or a known profit, but rather gives you an agreed-upon percentage of the profit if it occurs, and the loss is borne by the capital, then this is permissible.