Bretschneider's formula

In geometry, Bretschneider's formula is a mathematical expression for the area of a general quadrilateral.
It works on both convex and concave quadrilaterals, whether it is cyclic or not. The formula also works on crossed quadrilaterals provided that directed angles are used.

History

The German mathematician Carl Anton Bretschneider discovered the formula in 1842. The formula was also derived in the same year by the German mathematician Karl Georg Christian von Staudt.

Formulation

Bretschneider's formula is expressed as:
Here,,,, are the sides of the quadrilateral, is the semiperimeter, and and are any two opposite angles, since as long as directed angles are used so that or .

Proof

Denote the area of the quadrilateral by. Then we have
Therefore
The law of cosines implies that
because both sides equal the square of the length of the diagonal. This can be rewritten as
Adding this to the above formula for yields
Note that:
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 after taking the square root of both sides:
The second form is given by using the cosine half-angle identity
yielding
Emmanuel García has used the generalized half angle formulas to give an alternative proof.

Related formulae

Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle.
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 and to give