Rather than bet out on the turn where you are going to be called/raised by better hands anyway, I think it is better to induce a bluff with some weaker hands. Betting to prevent a bluff doesn't make much sense to me. Your pretty much pot committed by this time, although there is still a little room fold. It all depends on the read of the villian. He could be thinking he is pot committed here and he might as well bet it out where he has folding equity rather than having to call on the river. The reraise on the flop was almost a min raise. Perhaps a bigger reraise or a call was in order on the flop. He could have min raised with top pair and then felt he could call a raise that small to see what you would do on the turn and then after the check he was all in with top pair? He could of course have two pair here with kt,t9. But I doubt he has a tt or 99 he would have raised preflop. And I doubt he has just a pair of kings, too much action on the flop. But he could have the made straight on the flop with 87. He could also have some sort of pair and inside straight draw. That makes sense too.