What is the ruling on paying half the cost of car repairs after it was involved in an accident while a relative was authorized to sell it on my behalf, especially after I went back on the sale?

Your relative, who sold your car, acted as your agent, and his possession was one of trust. Therefore, he is not liable unless he committed an act of transgression or negligence. Since the one at fault is someone else, the liability falls upon the one at fault, not upon him or upon you. Your relative's waiving of half the fault is binding only upon him. You are not obligated to pay anything for that, unless you choose to donate to your relative as a form of assistance from you. You are also not obligated to sell the car if you do not wish to.