Your reaction doesn't have to be "normal" to be totally fine.

I guess you can confront him if you want it out in the open -- but honestly, the secrecy might kinda work for you? You don't have to hear about it ever and he can't spend too much time or money on her. It's really up to you.

If you come to a point where you'd like an affair of your own, you can re-evaluate whether to be more open about it or not. It's still up to you.