Bretschneider's formula

In geometry, Bretschneider's formula is a mathematical expression for the area of a general quadrilateral. It works on both convex and concave quadrilaterals, whether it is cyclic or not. The formula also works on crossed quadrilaterals provided that directed angles are used.

History

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.

Formulation

Bretschneider's formula is expressed as:

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

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

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

Proof

Denote the area of the quadrilateral by $K$ . Then we have

{\begin{aligned}K&={\frac {ad\sin \alpha }{2}}+{\frac {bc\sin \gamma }{2}}.\end{aligned}}

Therefore

2K=(ad)\sin \alpha +(bc)\sin \gamma .

4K^{2}=(ad)^{2}\sin ^{2}\alpha +(bc)^{2}\sin ^{2}\gamma +2abcd\sin \alpha \sin \gamma .

The law of cosines implies that

a^{2}+d^{2}-2ad\cos \alpha =b^{2}+c^{2}-2bc\cos \gamma ,

because both sides equal the square of the length of the diagonal $BD$ . This can be rewritten as

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

Adding this to the above formula for $4 K 2$ yields

{\begin{aligned}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{aligned}}

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

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\cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right).

Introducing the semiperimeter

s={\frac {a+b+c+d}{2}},

the above becomes

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

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

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

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

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

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

yielding

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

Emmanuel García has used the generalized half angle formulas to give an alternative proof. ^[1]

Related formulae

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 $e$ and $f$ to give^[2]^[3]

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

Notes

^ E. A. José García, Two Identities and their Consequences, MATINF, 6 (2020) 5-11. [1]
^ Coolidge, J. L. (1939). "A Historically Interesting Formula for the Area of a Quadrilateral". The American Mathematical Monthly. 46 (6): 345–347. doi:10.2307/2302891. JSTOR 2302891.
^ Hobson, E. W. (1918). A Treatise on Plane Trigonometry. Cambridge University Press. pp. 204–205.

References & further reading

Ayoub, Ayoub B. (2007). "Generalizations of Ptolemy and Brahmagupta Theorems". Mathematics and Computer Education. 41 (1). ISSN 0730-8639.
C. A. Bretschneider. Untersuchung der trigonometrischen Relationen des geradlinigen Viereckes. Archiv der Mathematik und Physik, Band 2, 1842, S. 225-261 (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 (online copy, German)

External links

Weisstein, Eric W. "Bretschneider's formula". MathWorld.
Bretschneider's formula at proofwiki.org
Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula at Dynamic Geometry Sketches, interactive geometry sketches.

[1] E. A. José García, Two Identities and their Consequences, MATINF, 6 (2020) 5-11. [1]

[2] Coolidge, J. L. (1939). "A Historically Interesting Formula for the Area of a Quadrilateral". The American Mathematical Monthly. 46 (6): 345–347. doi:10.2307/2302891. JSTOR 2302891.

[3] Hobson, E. W. (1918). A Treatise on Plane Trigonometry. Cambridge University Press. pp. 204–205.

[1]

[2]

[3]