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> why was every story in the ancient world written as a poem?
Uhm, it wasn't (the first counterexample I can think of is the Bible). But, poetry was popular, sure, because it is easier to remember and more pleasant to the ear. This was very important, in a time when learning by heart and public readings of books were widespread, given the scarcity of written copies.
> was there some reason it all had to rhyme
For the record, classical Greek and Latin poems do not rhyme but occasionally. The verses are composed according to the length of syllables, so that e.g. an ideal epic hexameter is composed of six feet of three syllables, long-short-short, minus the last one (it's actually much more complex but you get the idea).
> did people writing non-fiction, like Thucydides and Aristotle, write in verse?
Some did, like, dunno, Nicander, and Lucretius (who compared his treatise in verse to a sweetened medicine). The opposite also happened, take for example Lucian of Samosata's novels.
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Epic meme bruh, but this is a srs bread :^)