Bretschneider's formula
Suppose we have a quadrilateral with edges of length (in that order) and diagonals of length . Bretschneider's formula states that the area .
It can be derived with vector geometry.
Proof
Suppose a quadrilateral has sides such that and that the diagonals of the quadrilateral are and . The area of any such quadrilateral is .
Lagrange's Identity states that . Therefore:
Then if represent (and are thus the side lengths) while represent (and are thus the diagonal lengths), the area of a quadrilateral is: