Unfortunately, I think guys like my girlfriend's brother still believe that.
Girls very rarely ask guys out, because 99.9% of the time, they won't have to. Guys are expected to take all the risk and if they're not stupid and scared to death for no reason, they will.
So there's no reason that a girl will ever put herself in a situation where she could be rejected. Guys ask girls out, it's supposed to be common knowledge, except for hardcore feminists who wish that the world was how they want it to be.
The most a girl will do is to give signs, often very subtle to the point that a flat-out rejection will not be possible.
Smiling, talking to a guy, laughing at his jokes, giving him compliments, touching him lightly and playfully, etc.
After that, as a man, you need to escalate, you need to ask her out for something to happen.