Quadrilateral half-angle formulas relate half angles at opposite vertices of a convex quadrilateral
to its side lengths. Write 
, 
, 
,
and 
for the sides of 
,
and let 
be its semiperimeter.
With 
and 
denoting the interior angles at the corresponding
vertices, the law
of cosines applied to the diagonal 
gives 
.
Applying the half-angle formulas yields
 |  |  |
(1)
|
 |  |  |
(2)
|
Adding the areas of triangles 
and 
gives 
. Multiplying the two identities and
subtracting 
gives ![abcdcos^2[(A+C)/2]](/images/equations/QuadrilateralHalf-AngleFormulas/Inline21.svg)
.
This is José García's (2020) derivation of the angular form of Bretschneider's
formula,
![K^2=(s-a)(s-b)(s-c)(s-d)-abcdcos^2[(A+C)/2].](/images/equations/QuadrilateralHalf-AngleFormulas/NumberedEquation1.svg) |
(3)
|
For a cyclic quadrilateral, the opposite angles are supplementary, so the half-angle identities reduce to
 |  |  |
(4)
|
 |  |  |
(5)
|
Since the area in this case is 
, the two identities give Brahmagupta's
formula.
