Bretschneider's formula

In geometry, Bretschneider's formula is the following expression for the area of a general quadrilateral:

Here, a, b, c, d are the sides of the quadrilateral, s is the semiperimeter, and α and γ are two opposite angles.

Bretschneider's formula works on any quadrilateral, whether it is cyclic or not.

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.

Denote the area of the quadrilateral by K. Then we have

Therefore

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 4K² yields

Note that: ${\displaystyle \cos ^{2}{\frac {\alpha +\gamma }{2}}={\frac {1+\cos(\alpha +\gamma )}{2}}}$ (a trigonometric identity true for all ${\displaystyle {\frac {\alpha +\gamma }{2}}}$)

...
Wikipedia