On the turn, after he checks, the first question you should be asking is WHY? Your answer should be simple: He has you on a big overpair AND he has either: a. A smaller pair (my best guess); b. Or overcards (less likely, maybe he has AQ). If he's playing a whole lot of hands, maybe he has 87, but you really can't put him on this hand the way it was played both pre-flop and on the flop. Is A9s out of the question. No.

Checking on the turn is a surefire sign of weakenss. You three bet, now you are checking. Sure, it may lead to suspicion, but now chance you had of having him fold his small pair or his QK / AQ overcards is over. Put in another way, on the turn you are investing a small percentage of the pot to win a big one if he folds. On the river, you are putting a small percentage of the flop on a wild guess that will be right a very small portion of the time. Which would you rather have?

Given your situation, you have to bet on the turn. It's really your only shot at a fold. There are too many cards on the river that can help your opponent, at least from your perspective