Brahmagupta's Formula
Brahmagupta's Formula is a formula for determining the area of a cyclic quadrilateral given only the four side lengths.
Definition
Given a cyclic quadrilateral with side lengths ,
,
,
, the area
can be found as:
where is the semiperimeter of the quadrilateral.
Proof
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. Unknown error_msg)
Substituting results in
By the Law of Cosines,
.
, so a little rearranging gives
Similar formulas
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?
Problems
Intermediate
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)