Isang kuwadrilateral . Sa heometriya , ang pormula ni Bretschneider ay isang ekspresyong matematikal na ginagamit upang makuha ang kabuuang sukat ng pangkalahatang kuwadrilateral :
K
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
a
b
c
d
⋅
cos
2
(
α
+
γ
2
)
{\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\cdot \cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)}}}
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
1
2
a
b
c
d
[
1
+
cos
(
α
+
γ
)
]
.
{\displaystyle ={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{2}}abcd[1+\cos(\alpha +\gamma )]}}.}
Dito, ang a b c d s semi-perimetro , at ang α γ anggulong magkasalungat . Ang pormula ni Bretschneider ay maaaring gamitin sa anumang kuwadrilateral, maging ito ay sikliko o hindi.
Ang matematikong Aleman na si Carl Anton Bretschneider ang unang nakatuklas sa pormulang ito noong 1842. Ang pormula ay nakuha rin sa parehong taon ng dalubhasang Aleman na si Karl Georg Christian von Staudt .
Itukoy ang sukat ng kuwadrilateral bilang K
K
=
area of
△
A
D
B
+
area of
△
B
D
C
=
a
d
sin
α
2
+
b
c
sin
γ
2
.
{\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}}}
Samakatuwid, mabubuo natin ang dalawang tumbasan na
2
K
=
(
a
d
)
sin
α
+
(
b
c
)
sin
γ
.
{\displaystyle 2K=(ad)\sin \alpha +(bc)\sin \gamma .}
4
K
2
=
(
a
d
)
2
sin
2
α
+
(
b
c
)
2
sin
2
γ
+
2
a
b
c
d
sin
α
sin
γ
.
{\displaystyle 4K^{2}=(ad)^{2}\sin ^{2}\alpha +(bc)^{2}\sin ^{2}\gamma +2abcd\sin \alpha \sin \gamma .}
Ang batas ng mga cosine ay nagpapahiwatig na
a
2
+
d
2
−
2
a
d
cos
α
=
b
2
+
c
2
−
2
b
c
cos
γ
,
{\displaystyle a^{2}+d^{2}-2ad\cos \alpha =b^{2}+c^{2}-2bc\cos \gamma ,}
sapagkat ang magkabilang gilid ay magkatumbas sa magiging kuwadrado ng haba ng dayagonal BD
(
a
2
+
d
2
−
b
2
−
c
2
)
2
4
=
(
a
d
)
2
cos
2
α
+
(
b
c
)
2
cos
2
γ
−
2
a
b
c
d
cos
α
cos
γ
.
{\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 .}
Kung ipagsasama ito sa pormulang nasa itaas para sa 4K 2 , magiging
4
K
2
+
(
a
2
+
d
2
−
b
2
−
c
2
)
2
4
=
(
a
d
)
2
+
(
b
c
)
2
−
2
a
b
c
d
cos
(
α
+
γ
)
=
(
a
d
+
b
c
)
2
−
2
a
b
c
d
−
2
a
b
c
d
cos
(
α
+
γ
)
=
(
a
d
+
b
c
)
2
−
2
a
b
c
d
(
cos
(
α
+
γ
)
+
1
)
=
(
a
d
+
b
c
)
2
−
4
a
b
c
d
(
cos
(
α
+
γ
)
+
1
2
)
=
(
a
d
+
b
c
)
2
−
4
a
b
c
d
cos
2
(
α
+
γ
2
)
.
{\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}}}
Tandaan:
cos
2
α
+
γ
2
=
1
+
cos
(
α
+
γ
)
2
{\displaystyle \cos ^{2}{\frac {\alpha +\gamma }{2}}={\frac {1+\cos(\alpha +\gamma )}{2}}}
α
+
γ
2
{\displaystyle {\frac {\alpha +\gamma }{2}}}
Kung susundin ang parehong mga hakbang mula sa pormula ni Brahmagupta , maaari itong isulat bilang
16
K
2
=
(
a
+
b
+
c
−
d
)
(
a
+
b
−
c
+
d
)
(
a
−
b
+
c
+
d
)
(
−
a
+
b
+
c
+
d
)
−
16
a
b
c
d
cos
2
(
α
+
γ
2
)
.
{\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).}
Sa pagsasama ng semi-perimetrong
s
=
a
+
b
+
c
+
d
2
,
{\displaystyle s={\frac {a+b+c+d}{2}},}
ang pormulang nasa itaas ay magiging
16
K
2
=
16
(
s
−
d
)
(
s
−
c
)
(
s
−
b
)
(
s
−
a
)
−
16
a
b
c
d
cos
2
(
α
+
γ
2
)
{\displaystyle 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
)
−
a
b
c
d
cos
2
(
α
+
γ
2
)
{\displaystyle K^{2}=(s-a)(s-b)(s-c)(s-d)-abcd\cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)}
at ang pormula ni Bretschneider ay magiging sang-ayon pagkatapos makuha ang ugat ng kuwadrado ng magkabilang gilid:
K
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
a
b
c
d
⋅
cos
2
(
α
+
γ
2
)
{\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\cdot \cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)}}}
Ang pormula ni Bretschneider ay isang paglalahat ng pormula ni Brahmagupta para sa pagkuha ng sukat ng isang siklikong kuwadrilateral , na sa pangkalahatan ay naglalahat din sa pormula ni Heron para sa pagkuha ng sukat ng isang tatsulok .
Ang trigonometrikong pagsasaayos ng pormula ni Bretschneider para sa mga kuwadrilateral na hindi sikliko ay maaaring muling isulat na di-trigonometriko sa pamamagitan ng mga gilid at mga dayagonal na e f [ 1] [ 2]
K
=
1
4
4
e
2
f
2
−
(
b
2
+
d
2
−
a
2
−
c
2
)
2
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
1
4
(
a
c
+
b
d
+
e
f
)
(
a
c
+
b
d
−
e
f
)
.
{\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}}}
↑ 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.
↑ Hobson, E. W. (1918). A Treatise on Plane Trigonometry . Cambridge University Press. pp. 204–205
Ayoub, Ayoub B. (2007). "Generalizations of Ptolemy and Brahmagupta Theorems". Mathematics and Computer Education . 41 (1). ISSN 0730-8639 . {{cite journal }}: CS1 maint: date auto-translated (link )
CA Bretschneider. Untersuchung der trigonometrischen Relationen des geradlinigen Viereckes. Archiv der Mathematik und Physik, Band 2, 1842, S. 225-261 ( online na kopya, Aleman )
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 na kopya, Aleman ) 
![{\displaystyle ={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{2}}abcd[1+\cos(\alpha +\gamma )]}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/274a2fdb98ed6e71c98f8dee380c3e1318d9e4a7)
