You can not claim the amounts you pay except in the year you claim her as a dependent. So you each take turns claiming her , rotating each year. Making the payment directly just means it is not under gift tax rules. You could claim the medical deduction if the only reason you could not claim her was her income but that is not the case here because you can take turns.
Sometimes no one provides more than half of the support of a person. Instead, two or more persons, each of whom would be able to take the exemption but for the support test, together provide more than half of the person's support.
When this happens, you can agree that any one of you who individually provides more than 10% of the person's support, but only one, can claim an exemption for that person as a qualifying relative. Each of the others must sign a statement agreeing not to claim the exemption for that year. The person who claims the exemption must keep these signed statements for his or her records. A multiple support declaration identifying each of the others who agreed not to claim the exemption must be attached to the return of the person claiming the exemption. Form 2120 can be used for this purpose.
You can claim an exemption under a multiple support agreement for someone related to you or for someone who lived with you all year as a member of your household. Your mother would not need to live with any of you because parents are exempt from that.
You can not claim the amounts you pay except in the year you claim her as a dependent. So you each take turns claiming her , rotating each year.