6 added all test cases
source | link

MathematicaWolfram Language (Mathematica), 52 51 55 bytes

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

Try it online!

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.

enter image description here

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

Mathematica, 52 51 55 bytes

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

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.

enter image description here

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

Wolfram Language (Mathematica), 55 bytes

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

Try it online!

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.

enter image description here

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

5 Added 0&@@
source | link

Mathematica, 52 5151 55 bytes

Area@BooleanRegion[And0&@@Area@BooleanRegion[And,Simplex[{0{,}}~Join~#]&/@#]&

Shaved a few bytes off the trivial answer.

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

Try it online!

Replacing Area with DiscretizeRegion will show the intersection.

enter image description here

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

Mathematica, 52 51 bytes

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

Shaved a few bytes off the trivial answer.

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

Try it online!

Replacing Area with DiscretizeRegion will show the intersection.

enter image description here

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

-1 Byte thanks to JungHwan Min

Mathematica, 52 51 55 bytes

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

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.

enter image description here

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

4 added 233 characters in body
source | link

Mathematica, 52 51 bytes

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

Shaved a few bytes off the trivial answer.

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

Try it online!

Replacing Area with DiscretizeRegion will show the intersection.

enter image description here

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

-1 Byte thanks to JungHwan Min

Mathematica, 52 51 bytes

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

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.

enter image description here

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

-1 Byte thanks to JungHwan Min

Mathematica, 52 51 bytes

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

Shaved a few bytes off the trivial answer.

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

Try it online!

Replacing Area with DiscretizeRegion will show the intersection.

enter image description here

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

-1 Byte thanks to JungHwan Min

3 Added attribution, example, -1 byte
source | link
2 added 1 character in body
source | link
1
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