Partial solution to your problem: pastebin.com/bNYfTDvj From here you can let the angle vary, but the calculus gets iffy...
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Ooh! Can't wait to try!
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On further thought you only need to choose "opposite" triples of edges. That reduces it to 5 possibilities instead of 10.
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where opposite means they aren't all touching?
I have a hunch that at least one of the circumscribed points will be a corner of the minimum area polygon.
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