According to a voice actor at Valve, Half-Life 3 isn't being developed yet. And it's not just any voice actor - it's John Patrick Lowrie, a man who's featured in many of Valve's projects and is even married to Ellen McLain, a woman you may know better as the voice of GLaDOS.
According to Lowrie, the game isn't being made yet. That's not to say it's never going to happen (despite what the cynics say), just that it's not being made yet.
While the blog itself is a few years old, the comments section is always full of new material - including Lowrie giving the very quick and clear response of: "Sorry, I’m afraid that the opposite is true. HL3 is not being worked on at this time as far as I know."
As it turns out, the primary reason for the delay is to do with motion capture:
"Here is the biggest challenge with bringing out HL3: the big thing now with FPSs is motion capture, or mo-cap," Lowrie says. "One of the great things about HL2 is that all of the characters that you meet actually look at you when they talk to you no matter where you go or stand. With mo-cap you can’t do that, at least not yet. Once you film the actor doing something and capture that motion, that’s what the character is going to do. This works great in movies, but when you make something interactive it gets way less interactive with mo-cap. So that’s one of the things they’re working on."
"... As far as I know they are not developing HL3 now for several reasons, among them the mo-cap issue. Sorry for any confusion. What they might decide in the future depends on lots of different factors. I hope they do, personally, but it has to make sense for them."