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| [[File:Tetragon measures.svg|thumb|230px|A quadrilateral.]]
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| In [[geometry]], '''Bretschneider's formula''' is the following expression for the [[area]] of a general convex [[quadrilateral]]:
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| :<math> K = \sqrt {(s-a)(s-b)(s-c)(s-d) - abcd \cdot \cos^2 \left(\frac{\alpha + \gamma}{2}\right)}.</math>
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| Here, ''a'', ''b'', ''c'', ''d'' are the sides of the quadrilateral, ''s'' is the [[semiperimeter]], and <math>\alpha \,</math> and <math>\gamma \,</math> are two opposite angles.
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| Bretschneider's formula works on any convex quadrilateral, whether it is [[cyclic quadrilateral|cyclic]] or not.
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| The German mathematician [[Carl Anton Bretschneider]] discovered the formula in 1842. The formula was also derived in the same year by the German mathematician [[Karl Georg Christian von Staudt]].
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| == Proof of Bretschneider's formula ==
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| Denote the area of the quadrilateral by ''K''. Then we have
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| :<math> \begin{align} K &= \text{area of } \triangle ADB + \text{area of } \triangle BDC \\
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| &= \frac{a d \sin \alpha}{2} + \frac{b c \sin \gamma}{2}.
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| \end{align} </math>
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| Therefore
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| :<math> 4K^2 = (ad)^2 \sin^2 \alpha + (bc)^2 \sin^2 \gamma + 2abcd \sin \alpha \sin \gamma. \, </math>
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| The [[Law of Cosines]] implies that
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| :<math> a^2 + d^2 -2ad \cos \alpha = b^2 + c^2 -2bc \cos \gamma, \, </math>
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| because both sides equal the square of the length of the diagonal ''BD''. This can be rewritten as
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| :<math>\frac{(a^2 + d^2 - b^2 - c^2)^2}{4} = (ad)^2 \cos^2 \alpha +(bc)^2 \cos^2 \gamma -2 abcd \cos \alpha \cos \gamma. \,</math> | |
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| Adding this to the above formula for <math>4K^2</math> yields
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| :<math> \begin{align} 4K^2 + \frac{(a^2 + d^2 - b^2 - c^2)^2}{4} &= (ad)^2 + (bc)^2 - 2abcd \cos (\alpha + \gamma) \\
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| &= (ad + bc)^2 - 4abcd \cos^2 \left(\frac{\alpha + \gamma}{2}\right).
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| \end{align} </math>
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| Following the same steps as in [[Brahmagupta's formula#Trigonometric proof|Brahmagupta's formula]], this can be written as
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| :<math>16K^2 = (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d) - 16abcd \cos^2 \left(\frac{\alpha + \gamma}{2}\right).</math>
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| Introducing the semiperimeter
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| :<math>s = \frac{a+b+c+d}{2},</math>
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| the above becomes
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| :<math>16K^2 = 16(s-a)(s-b)(s-c)(s-d) - 16abcd \cos^2 \left(\frac{\alpha + \gamma}{2}\right)</math>
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| and Bretschneider's formula follows.
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| == Related formulas ==
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| '''Bretschneider's formula''' generalizes [[Brahmagupta's formula]] for the area of a [[cyclic quadrilateral]], which in turn generalizes [[Heron's formula]] for the area of a [[triangle]].
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| == External links ==
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| * {{MathWorld|urlname=BretschneidersFormula|title=Bretschneider's formula}}
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| [[Category:Quadrilaterals]]
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| [[Category:Area]]
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| [[Category:Articles containing proofs]]
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Oscar is how he's called and he completely loves this name. For years he's been working as a receptionist. To collect badges is what her family members and her appreciate. Minnesota is where he's been living for many years.
my page; jewelrycase.co.kr