Difference between revisions of "Brahmagupta's Formula"
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− | '''Brahmagupta's | + | '''Brahmagupta's Formula''' is a [[formula]] for determining the [[area]] of a [[cyclic quadrilateral]] given only the four side [[length]]s, given as follows: <cmath>[ABCD] = \sqrt{(s-a)(s-b)(s-c)(s-d)}</cmath> where <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> are the four side lengths and <math>s = \frac{a+b+c+d}{2}</math>. |
+ | ==Proofs== | ||
+ | If we draw <math>AC</math>, we find that <math>[ABCD]=\frac{ab\sin B}{2}+\frac{cd\sin D}{2}=\frac{ab\sin B+cd\sin D}{2}</math>. Since <math>B+D=180^\circ</math>, <math>\sin B=\sin D</math>. Hence, <math>[ABCD]=\frac{\sin B(ab+cd)}{2}</math>. Multiplying by 2 and squaring, we get: | ||
+ | <cmath>4[ABCD]^2=\sin^2 B(ab+cd)^2</cmath> | ||
+ | Substituting <math>\sin^2B=1-\cos^2B</math> results in | ||
+ | <cmath>4[ABCD]^2=(1-\cos^2B)(ab+cd)^2=(ab+cd)^2-\cos^2B(ab+cd)^2</cmath> | ||
+ | By the [[Law of Cosines]], <math>a^2+b^2-2ab\cos B=c^2+d^2-2cd\cos D</math>. <math>\cos B=-\cos D</math>, so a little rearranging gives | ||
+ | <cmath>2\cos B(ab+cd)=a^2+b^2-c^2-d^2</cmath> | ||
+ | <cmath>4[ABCD]^2=(ab+cd)^2-\frac{1}{4}(a^2+b^2-c^2-d^2)^2</cmath> | ||
+ | <cmath>16[ABCD]^2=4(ab+cd)^2-(a^2+b^2-c^2-d^2)^2</cmath> | ||
+ | <cmath>16[ABCD]^2=(2(ab+cd)+(a^2+b^2-c^2-d^2))(2(ab+cd)-(a^2+b^2-c^2-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+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>[ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)}</cmath> | ||
− | == | + | == 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 amnado | |
− | <math> | + | Brahmagupta's formula reduces to [[Heron's formula]] by setting the side length <math>{d}=0</math>. |
− | + | A similar formula which Brahmagupta derived for the area of a general quadrilateral is | |
+ | <cmath>[ABCD]^2=(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\left({\frac{B+D}{2}}\right)</cmath> | ||
+ | <cmath>[ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\left({\frac{B+D}{2}}\right)}</cmath> | ||
+ | where <math>s=\frac{a+b+c+d}{2}</math> is the [[semiperimeter]] of the quadrilateral. What happens when the quadrilateral is cyclic? | ||
− | == | + | == 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 18:31, 3 July 2023
Brahmagupta's Formula is a formula for determining the area of a cyclic quadrilateral given only the four side lengths, given as follows: where
,
,
,
are the four side lengths and
.
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 amnado
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)