I don't think the half pot bet was enough of a show of strength for you to fold. It was a feeler bet. "Is there an A out there?" It also might push flush draws out of the pot. Although at these limits, anything is possible.

Another use of this bet is to get some money into the pot for the trips. But sooner or later they need to show some strength to maximize the value of their hand and you can abandon yours then if you sense that strength and that they are not bluffing.

I notice from your post that you don't mention your 'read' on this player, which is more telling than any sage advice on particular strategies for starting hands.

My personal poker style is evolving from one of starting hand rules to getting a sense of the relative value of my hand (no matter what it is) and what my opponents are doing based upon my read and their betting patterns. I learnt this through losing AA, AKs, KK hands.

There are many ways to play this hand. A pot sized raise, for example, might have won the pot on the spot. And even if it lost it might have altered your table image to suit you in some future play. I don't mind losing small pots if it encourages people to be looser with my stronger hands in the future.

Good luck.