Given a cyclic quadrilateral with side lengths , , , , the area can be found as:
where is the semiperimeter of the quadrilateral.
If we draw , we find that . Since , . Hence, . Multiplying by 2 and squaring, we get:
\[4[ABCD]}^2=\sin^2 B(ab+cd)^2\] (Error compiling LaTeX. ! Extra }, or forgotten $.)
Substituting results in By the Law of Cosines, . , so a little rearranging gives
Bretschneider's formula gives a formula for the area of a non-cyclic quadrilateral given only the side lengths; applying Ptolemy's Theorem to Bretschneider's formula reduces it to Brahmagupta's formula.
Brahmagupta's formula reduces to Heron's formula by setting the side length .
A similar formula which Brahmagupta derived for the area of a general quadrilateral is where is the semiperimeter of the quadrilateral. What happens when the quadrilateral is cyclic?
- is a cyclic quadrilateral that has an inscribed circle. The diagonals of intersect at . If and then the area of the inscribed circle of can be expressed as , where and are relatively prime positive integers. Determine . (Source)