Difference between revisions of "Brahmagupta's Formula"
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<cmath>[ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2({\frac{B+D}{2}}})</cmath> | <cmath>[ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2({\frac{B+D}{2}}})</cmath> | ||
where <math>s=\frac{a+b+c+d}{2}</math> is the [[semiperimeter]] of the quadrilateral. What happens when the quadrilateral is cyclic? | where <math>s=\frac{a+b+c+d}{2}</math> is the [[semiperimeter]] of the quadrilateral. What happens when the quadrilateral is cyclic? | ||
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+ | == Problems == | ||
+ | === Intermediate === | ||
+ | *<math>ABCD</math> is a cyclic quadrilateral that has an inscribed circle. The diagonals of <math>ABCD</math> intersect at <math>P</math>. If <math>AB = 1, CD = 4,</math> and <math>BP : DP = 3 : 8,</math> then the area of the inscribed circle of <math>ABCD</math> can be expressed as <math>\frac{p\pi}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Determine <math>p + q</math>. ([[Mock AIME 3 Pre 2005 Problems/Problem 7|Source]]) | ||
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[[Category:Geometry]] | [[Category:Geometry]] | ||
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[[Category:Theorems]] | [[Category:Theorems]] |
Revision as of 15:16, 3 April 2012
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)