Here, a, b, c, d are the sides of the quadrilateral, s is the semiperimeter, and and are two opposite angles.
Bretschneider's formula works on any convex quadrilateral, whether it is cyclic or not.
Proof of Bretschneider's formula
Denote the area of the quadrilateral by K. Then we have
The Law of Cosines implies that
because both sides equal the square of the length of the diagonal BD. This can be rewritten as
Adding this to the above formula for yields
Following the same steps as in Brahmagupta's formula, this can be written as
Introducing the semiperimeter
the above becomes
and Bretschneider's formula follows.
The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals p and q to give
- J. L. Coolidge, "A historically interesting formula for the area of a quadrilateral", American Mathematical Monthly, 46 (1939) 345–347.