Bretschneider's formula

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.

Bretschneider's formula is expressed as:

:$K\; =\; \backslash sqrt\; \{(s-a)(s-b)(s-c)(s-d)\; -\; abcd\; \backslash cdot\; \backslash cos^2\; \backslash left(\backslash frac\{\backslash alpha\; +\; \backslash gamma\}\{2\}\backslash right)\}$

::$=\; \backslash sqrt\{(s-a)(s-b)(s-c)(s-d)\; -\; \backslash tfrac\{1\}\{2\}\; abcd\; [\; 1\; +\; \backslash cos\; (\backslash alpha+\; \backslash gamma)\; ]\}\; .$

Here, {{math|a}}, {{math|b}}, {{math|c}}, {{math|d}} are the sides of the quadrilateral, {{math|s}} is the semiperimeter, and {{math|α}} and {{math|γ}} are any two opposite angles, since $\backslash cos\; (\backslash alpha+\; \backslash gamma)\; =\; \backslash cos\; (\backslash beta+\; \backslash delta)$ as long as directed angles are used so that $\backslash alpha+\backslash beta+\backslash gamma+\backslash delta=360^\{\backslash circ\}$ or $\backslash alpha+\backslash beta+\backslash gamma+\backslash delta=720^\{\backslash circ\}$ (when the quadrilateral is crossed).

Denote the area of the quadrilateral by {{math|K}}. Then we have

:

Therefore

:$2K=\; (ad)\; \backslash sin\; \backslash alpha\; +\; (bc)\; \backslash sin\; \backslash gamma.$

:$4K^2\; =\; (ad)^2\; \backslash sin^2\; \backslash alpha\; +\; (bc)^2\; \backslash sin^2\; \backslash gamma\; +\; 2abcd\; \backslash sin\; \backslash alpha\; \backslash sin\; \backslash gamma.$

The law of cosines implies that

:$a^2\; +\; d^2\; -2ad\; \backslash cos\; \backslash alpha\; =\; b^2\; +\; c^2\; -2bc\; \backslash cos\; \backslash gamma,$

because both sides equal the square of the length of the diagonal {{math|BD}}. This can be rewritten as

:$\backslash frac\{(a^2\; +\; d^2\; -\; b^2\; -\; c^2)^2\}\{4\}\; =\; (ad)^2\; \backslash cos^2\; \backslash alpha\; +(bc)^2\; \backslash cos^2\; \backslash gamma\; -2\; abcd\; \backslash cos\; \backslash alpha\; \backslash cos\; \backslash gamma.$

Adding this to the above formula for {{math|4K{{sup|2}}}} yields

:$\backslash begin\{align\}$

4K^2 + \frac{(a^2 + d^2 - b^2 - c^2)^2}{4} &= (ad)^2 + (bc)^2 - 2abcd \cos (\alpha + \gamma) \\

&= (ad+bc)^2-2abcd-2abcd\cos(\alpha+\gamma) \\

&= (ad+bc)^2 - 2abcd(\cos(\alpha+\gamma)+1) \\

&= (ad+bc)^2 - 4abcd\left(\frac{\cos(\alpha+\gamma)+1}{2}\right) \\

&= (ad + bc)^2 - 4abcd \cos^2 \left(\frac{\alpha + \gamma}{2}\right).

\end{align}

Note that: $\backslash cos^2\backslash frac\{\backslash alpha+\backslash gamma\}\{2\}\; =\; \backslash frac\{1+\backslash cos(\backslash alpha+\backslash gamma)\}\{2\}$ (a trigonometric identity true for all $\backslash frac\{\backslash alpha+\backslash gamma\}\{2\}$)

Following the same steps as in Brahmagupta's formula, this can be written as

:$16K^2\; =\; (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)\; -\; 16abcd\; \backslash cos^2\; \backslash left(\backslash frac\{\backslash alpha\; +\; \backslash gamma\}\{2\}\backslash right).$

Introducing the semiperimeter

:$s\; =\; \backslash frac\{a+b+c+d\}\{2\},$

the above becomes

:$16K^2\; =\; 16(s-d)(s-c)(s-b)(s-a)\; -\; 16abcd\; \backslash cos^2\; \backslash left(\backslash frac\{\backslash alpha\; +\; \backslash gamma\}\{2\}\backslash right)$

:$K^2\; =\; (s-a)(s-b)(s-c)(s-d)\; -\; abcd\; \backslash cos^2\; \backslash left(\backslash frac\{\backslash alpha\; +\; \backslash gamma\}\{2\}\backslash right)$

and Bretschneider's formula follows after taking the square root of both sides:

:$K\; =\; \backslash sqrt\; \{(s-a)(s-b)(s-c)(s-d)\; -\; abcd\; \backslash cdot\; \backslash cos^2\; \backslash left(\backslash frac\{\backslash alpha\; +\; \backslash gamma\}\{2\}\backslash right)\}$

The second form is given by using the cosine half-angle identity

:$\backslash cos^2\; \backslash left(\backslash frac\{\backslash alpha\; +\; \backslash gamma\}\{2\}\backslash right)\; =\; \backslash frac\; \{1\; +\; \backslash cos\; \backslash left(\backslash alpha\; +\; \backslash gamma\backslash right)\}\{2\},$

yielding

:$K\; =\; \backslash sqrt\{(s-a)(s-b)(s-c)(s-d)\; -\; \backslash tfrac\{1\}\{2\}\; abcd\; [\; 1\; +\; \backslash cos\; (\backslash alpha+\; \backslash gamma)\; ]\}\; .$

Emmanuel García has used the generalized half angle formulas to give an alternative proof. E. A. José García, Two Identities and their Consequences, MATINF, 6 (2020) 5-11. [http://matinf.upit.ro/MATINF6/index.html?fbclid=IwAR1l1rO0oOyEFW8UHuh5Mlt3gv_w8ibykYZGbVM7x2EKG7t3rHSc-vSMH7A#p=1]

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.

The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals {{math|e}} and {{math|f}} to give{{cite journal |last1=Coolidge |first1=J. L. |title=A Historically Interesting Formula for the Area of a Quadrilateral |journal=The American Mathematical Monthly |date=1939 |volume=46 |issue=6 |pages=345–347 |doi=10.2307/2302891|jstor=2302891 }}{{cite book|author1-link=E. W. Hobson|first1=E. W.|last1=Hobson|title=A Treatise on Plane Trigonometry|publisher=Cambridge University Press|date= 1918|pages= 204–205|url=https://archive.org/details/treatiseonplanet00hobs/page/n7/mode/2up}}

:$$

\begin{align}

K &=\tfrac{1}{4}\sqrt{4e^2f^2-(b^2+d^2-a^2-c^2)^2} \\

&=\sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{4}((ac+bd)^2-e^2f^2)} \\

&=\sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{4}(ac+bd+ef)(ac+bd-ef)} \\

\end{align}

{{reflist}}

• {{cite journal|first1=Ayoub B. |last1=Ayoub|title=Generalizations of Ptolemy and Brahmagupta Theorems|journal= Mathematics and Computer Education|volume= 41|issue= 1|date=2007|issn=0730-8639}}
• C. A. Bretschneider. Untersuchung der trigonometrischen Relationen des geradlinigen Viereckes. Archiv der Mathematik und Physik, Band 2, 1842, S. 225-261 ([https://books.google.com/books?id=7vxZAAAAYAAJ&pg=PA225 online copy, German])
• F. Strehlke: Zwei neue Sätze vom ebenen und sphärischen Viereck und Umkehrung des Ptolemaischen Lehrsatzes. Archiv der Mathematik und Physik, Band 2, 1842, S. 323-326 ([https://books.google.com/books?id=7vxZAAAAYAAJ&pg=PA323 online copy, German])

• {{MathWorld|urlname=BretschneidersFormula|title=Bretschneider's formula}}
• [https://proofwiki.org/wiki/Bretschneider%27s_Formula Bretschneider's formula] at proofwiki.org
• [https://dynamicmathematicslearning.com/brahmagupta-crossed-cyclic.html Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches], interactive geometry sketches.

Category:Theorems about quadrilaterals

Category:Area

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