Difference between revisions of "Brahmagupta's Formula"
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<cmath>16[ABCD]^2=(a^2+2ab+b^2-c^2+2cd-d^2)(-a^2+2ab-b^2+c^2+2cd+d^2)</cmath> | <cmath>16[ABCD]^2=(a^2+2ab+b^2-c^2+2cd-d^2)(-a^2+2ab-b^2+c^2+2cd+d^2)</cmath> | ||
<cmath>16[ABCD]^2=((a+b)^2-(c-d)^2)((c+d)^2-(a-b)^2)</cmath> | <cmath>16[ABCD]^2=((a+b)^2-(c-d)^2)((c+d)^2-(a-b)^2)</cmath> | ||
− | <cmath>16[ABCD]^2=(a+b+c-d)(a+b-c+d)(c+d+a-b)(c+d-b | + | <cmath>16[ABCD]^2=(a+b+c-d)(a+b-c+d)(c+d+a-b)(c+d-a+b)</cmath> |
<cmath>16[ABCD]^2=16(s-a)(s-b)(s-c)(s-d)</cmath> | <cmath>16[ABCD]^2=16(s-a)(s-b)(s-c)(s-d)</cmath> | ||
<cmath>[ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)}</cmath> | <cmath>[ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)}</cmath> |
Revision as of 14:23, 30 March 2018
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.
Proofs
If we draw , we find that
. Since
,
. Hence,
. Multiplying by 2 and squaring, we get:
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)