You are very fortunate because you paid by credit card. You should immediately contact your credit card company and dispute the charge. You should be able to reverse some or all of it because you paid for services that you did not recieve.
If the dentist filed for bankruptcy and listed you as a creditor, you should have received an official "notice of commencement" from the bankruptcy court.
The bankruptcy court appoints a Chapter 7 trustee to review his case and examine him. If he did have some kind of buy-sell arrangement with another dentist, the trustee will certainly investigate that.
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I agree with the other answer that you should try to cancel the charge with the credit card company. You can also call the bankruptcy clerk of court and ask them if the orthodontist (or his office) have filed bankruptcy. If he/it has, find out who is the trustee and ask them if there is going to be any money available to creditors. If there is, you can file a proof of claim and get at least something back. Good luck!.