Bretschneider's formula
Revision as of 03:51, 12 February 2021 by Twod horse (talk | contribs)
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: