I'm sure my high school English teacher meant well when she uttered this definition:
Protagonists are striving for something. Pro = for.
Antagonists are striving against something. Anti = against.
She only got it half right. As we saw yesterday, the pro in protagonist comes from proto, meaning first. The protagonist is the first actor, the deuteragonist is the second, the tritagonist is the third. And they're all working against the backdrop of a Greek chorus. So that's four classes of humans on the stage (though more than four physical bodies -- if memory serves, a Greek chorus had between a dozen and fifty people).
The protagonist was not originally defined as the character with the goal. It was defined as the character on stage with the chorus all the time. This meant, by extension, that the protagonist's character would be most central to the story. Otherwise, there would be no need to have him on stage all the time. The deuteragonist and tritagonist, with the many roles they fulfilled, could be on or off stage as the scene required. My understanding is that they generally remained on stage and changed their props in full view of the audience.
Where is the antagonist on the stage?
If you've asked this question, you've assumed that the antagonist is a character. And the thing is, maybe the antagonist is a character, and maybe not. Maybe the antagonist is different characters at different times. Maybe the protagonist gets a turn at being the antagonist. Maybe the antagonist is something intangible like the weather, not a character at all.
I think this begins to snap into focus if we stop talking about "the antagonist" and start talking about "antagonism." What is the source of antagonism in a particular story? Are there multiple sources or a single source?
In some stories, the antagonism will be embodied in a single identifiable character: the sheriff wears the white hat, and the outlaw wears the black hat, and good and bad collide over the course of the story until justice prevails and the noose is tied. It's very easy to understand this story model and apply literary definitions to the various component parts. Maybe that's ultimately a source of confusion, though, if we come to think of protagonists as always the "good guys" and antagonists as always the "bad guys."
In some stories, particularly in ensemble stories, the antagonism will be embodied in different characters at different times. Alliances are built and shift and fall. The protagonist gathers different pieces of the puzzle from different sources. Conflict can be very subtle and abstract in these stories -- think of The Big Chill, for example, which is built in layers of interaction between the protagonist (Kevin Kline) and his college friends. We might identify the conflict as Kline's struggle to cling to his nostalgic idealism in the face of what they've all become. The antagonism is released through different characters at different times: Goldblum's bitterness, Tilly's objectivity, Close's earth-mothering all serve to reveal different nuances of the conflict.
In some stories, the antagonism will not originate with a character at all. In Castaway, for example, the classic human versus nature struggle, there is only one character on screen for most of the story. There is no other body, so the antagonism quite literally can never be embodied. Does this mean the Tom Hanks character has no struggles? Of course not. But his struggles originate with something other than another human being. The antagonism is revealed and shaped through rain, coconuts, volleyballs, and so on.
Keep in mind that any of the four classes of people on stage can provide a source of antagonism. The deuteragonist mother may be loving and supportive in one scene, and then cast the protagonist into the wilderness in the next scene. The chorus may sing a character's praises in one scene, and lament his foolishness in the next.
So when we say that conflict is defined as the interaction of the protagonist and the antagonist, we're really talking about notions that came into existence through different channels and, in some ways, are independent story functions. Protagonist originates in character. Antagonist originates in conflict.
When you're doing global, big-picture thinking about your story, think about the source of the antagonism. Have you slotted your characters into "good" and "bad" roles, and missed out on potential ways to twist the conflict? Have you ignored the possibility of antagonism from non-human sources?