Bretschneider's formula

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

${\displaystyle {\begin{aligned}K&={\text{area of }}\triangle ADB+{\text{area of }}\triangle BDC\\&={\frac {ad\sin \alpha }{2}}+{\frac {bc\sin \gamma }{2}}.\end{aligned}}}$

Therefore

${\displaystyle 2K=(ad)\sin \alpha +(bc)\sin \gamma .}$

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

The law of cosines implies that

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

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

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

Adding this to the above formula for 4K² yields

${\displaystyle {\begin{aligned}4K^{2}+{\frac {(a^{2}+d^{2}-b^{2}-c^{2})^{2}}{4}}&=(ad)^{2}+(bc)^{2}-2abcd\cos(\alpha +\gamma )\\&=(ad+bc)^{2}-2abcd-2abcd\cos(\alpha +\gamma )\\&=(ad+bc)^{2}-2abcd(\cos(\alpha +\gamma )+1)\\&=(ad+bc)^{2}-4abcd\left({\frac {\cos(\alpha +\gamma )+1}{2}}\right)\\&=(ad+bc)^{2}-4abcd\cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right).\end{aligned}}}$

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

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

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

Introducing the semiperimeter

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

the above becomes

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

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

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

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

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[1][2]

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

• Ayoub B. Ayoub: Generalizations of Ptolemy and Brahmagupta Theorems. Mathematics and Computer Education, Volume 41, Number 1, 2007, ISSN 0730-8639
• E. W. Hobson: A Treatise on Plane Trigonometry. Cambridge University Press, 1918, pp. 204–205 (online copy)
• 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)

• Weisstein, Eric W. "Bretschneider's formula". MathWorld.
• Bretschneider's formula at proofwiki.org