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[parent] proof of Brahmagupta's formula (Proof)

We shall prove that the area of a cyclic quadrilateral with sides $ p,q,r,s$ is given by

$\displaystyle \sqrt{(T-p)(T-q)(T-r)(T-s)}$
where $ T = \frac{p+q+r+s}{2}.$
\includegraphics{quadcyclic.eps}

Area of the cyclic quadrilateral = Area of $ \triangle ADB +$ Area of $ \triangle BDC.$

$\displaystyle = \frac{1}{2}pq\sin A + \frac{1}{2}rs\sin C$
But since $ ABCD$ is a cyclic quadrilateral, $ \angle DAB = 180^\circ - \angle DCB.$ Hence $ \sin A = \sin C.$ Therefore area now is

$\displaystyle Area = \frac{1}{2}pq\sin A + \frac{1}{2}rs\sin A$

$\displaystyle (Area)^2 = \frac{1}{4}\sin^2 A (pq + rs)^2$

$\displaystyle 4(Area)^2 = (1 - \cos^2 A)(pq + rs)^2$

$\displaystyle 4(Area)^2 = (pq + rs)^2 - cos^2 A (pq + rs)^2$
Applying cosines law for $ \triangle ADB$ and $ \triangle BDC$ and equating the expressions for side $ DB,$ we have

$\displaystyle p^2 + q^2 - 2pq\cos A = r^2 + s^2 - 2rs\cos C$
Substituting $ \cos C = -\cos A$ (since angles $ A$ and $ C$ are suppplementary) and rearranging, we have

$\displaystyle 2\cos A (pq + rs) = p^2 + q^2 - r^2 - s^2$
substituting this in the equation for area,

$\displaystyle 4(Area)^2 = (pq + rs)^2 - \frac{1}{4}(p^2 + q^2 - r^2 - s^2)^2$

$\displaystyle 16(Area)^2 = 4(pq + rs)^2 - (p^2 + q^2 - r^2 - s^2)^2$
which is of the form $ a^2-b^2$ and hence can be written in the form $ (a+b)(a-b)$ as

$\displaystyle (2(pq + rs) + p^2 + q^2 -r^2 - s^2)(2(pq + rs) - p^2 - q^2 + r^2 +s^2)$

$\displaystyle = ( (p+q)^2 - (r-s)^2 )( (r+s)^2 - (p-q)^2 ) $

$\displaystyle = (p+q+r-s)(p+q+s-r)(p+r+s-q)(q+r+s-p)$
Introducing $ T = \frac{p+q+r+s}{2},$

$\displaystyle 16(Area)^2 = 16(T-p)(T-q)(T-r)(T-s)$
Taking square root, we get

$\displaystyle Area = \sqrt{(T-p)(T-q)(T-r)(T-s)}$



"proof of Brahmagupta's formula" is owned by giri.
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Cross-references: square root, equation, angles, expressions, cosines law, sides, cyclic quadrilateral, area

This is version 3 of proof of Brahmagupta's formula, born on 2002-11-14, modified 2002-11-14.
Object id is 3594, canonical name is ProofOfBrahmaguptasFormula.
Accessed 6594 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
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