Intersection of two triangles
Wolfram Language (Mathematica), 55 bytes

0&amp;@@Area@BooleanRegion[And,Simplex[{0{,}}~Join~#]&amp;/@#]&amp;


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Shaved a few bytes off the trivial answer.

%@{{{5, 1}, {1, 3}}, {{3, 4}, {4, -3}}} yields 46375/10296 or 4.504176379


Replacing Area with DiscretizeRegion will show the intersection.



By the way, this will work with any simplexes, not just triangles.

-1 Byte thanks to JungHwan Min

@user202729's suggestion added 4 bytes but makes it yield 0 for degenerate triangles