في الهندسة الرياضية، تقوم صيغة براهماگوپتا بإيجاد مساحة أي رباعي أضلاع بواسطة طول أضلاعه وقياس بعض زواياه.
بشكلها الأكثر شيوعاً تقوم المعادلة بحساب معادلة رباعي الأضلاع المحصور ضمن دائرة (رباعي دائري).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
الصيغة البسيطة
أبسط صيغة لصيغة براهماگوپتا هي الصيغة التي تعطى في الرباعي الدائري الذي أطوال أضلاعهa, b, c, d على الشكل التالي:
![{\displaystyle {\sqrt {(s-a)(s-b)(s-c)(s-d)}}}](https://www.marefa.org/api/rest_v1/media/math/render/svg/c5bb6f53fd4ca88e345a311cabd057a894800c08)
حيث s تعطى بالعلاقة:
![{\displaystyle s-a={\frac {-a+b+c+d}{2}}}](https://www.marefa.org/api/rest_v1/media/math/render/svg/e529b275d6af3ea874dc79cb2eeb859d7ef1414b)
![{\displaystyle s-b={\frac {a-b+c+d}{2}}}](https://www.marefa.org/api/rest_v1/media/math/render/svg/b05d3c42a848e933f6fa4a7758c1c32c1814a72c)
![{\displaystyle s-c={\frac {a+b-c+d}{2}}}](https://www.marefa.org/api/rest_v1/media/math/render/svg/3268c936d8013dd38fd72d8c912d38ba602b003c)
![{\displaystyle s-d={\frac {a+b+c-d}{2}}}](https://www.marefa.org/api/rest_v1/media/math/render/svg/5188b7e4b293f082fb343fe9a74a3075f5559f0b)
وهي تعميم لمعادلة هيرون لحساب مساحة المثلث.
![{\displaystyle K={\frac {\sqrt {(a^{2}+b^{2}+c^{2}+d^{2})^{2}+8abcd-2(a^{4}+b^{4}+c^{4}+d^{4})}}{4}}\cdot }](https://www.marefa.org/api/rest_v1/media/math/render/svg/2948bf13bd61cb8fd6e01843908bff70c1e7a93c)
اثبات صيغة براهماگوپتا
مخطط مرجعي
Here we use the notations in the figure to the right. Area of the cyclic quadrilateral = Area of
+ Area of
![{\displaystyle ={\frac {1}{2}}pq\sin A+{\frac {1}{2}}rs\sin C.}](https://www.marefa.org/api/rest_v1/media/math/render/svg/5fa3ac96481179a47ac6651e138bfc0f9c9f84b8)
But since
is a cyclic quadrilateral,
Hence
Therefore
![{\displaystyle {\mbox{Area}}={\frac {1}{2}}pq\sin A+{\frac {1}{2}}rs\sin A}](https://www.marefa.org/api/rest_v1/media/math/render/svg/e3cf7527a016705d4acd680fb5a28d72fa3ad3b5)
![{\displaystyle ({\mbox{Area}})^{2}={\frac {1}{4}}\sin ^{2}A(pq+rs)^{2}}](https://www.marefa.org/api/rest_v1/media/math/render/svg/3fb18f2f15d4b32fa6f7c9ee5d88096f831e2152)
![{\displaystyle 4({\mbox{Area}})^{2}=(1-\cos ^{2}A)(pq+rs)^{2}=(pq+rs)^{2}-\cos ^{2}A(pq+rs)^{2}.\,}](https://www.marefa.org/api/rest_v1/media/math/render/svg/e87a57119f6bc95a5fb6e4bfe77765d7911cb7f0)
Solving for common side DB, in
ADB and
BDC, the law of cosines gives
![{\displaystyle p^{2}+q^{2}-2pq\cos A=r^{2}+s^{2}-2rs\cos C.\,}](https://www.marefa.org/api/rest_v1/media/math/render/svg/10a529d79b8839a12344fba66f2699bc90f26911)
Substituting
(since angles
and
are supplementary) and rearranging, we have
![{\displaystyle 2\cos A(pq+rs)=p^{2}+q^{2}-r^{2}-s^{2}.\,}](https://www.marefa.org/api/rest_v1/media/math/render/svg/b60cee5c8b593789b02866684f12b8dd6cbbaae5)
Substituting this in the equation for the area,
![{\displaystyle 4({\mbox{Area}})^{2}=(pq+rs)^{2}-{\frac {1}{4}}(p^{2}+q^{2}-r^{2}-s^{2})^{2}}](https://www.marefa.org/api/rest_v1/media/math/render/svg/554aa349d97cf6912430e1aec68a93bb5ed100cb)
![{\displaystyle 16({\mbox{Area}})^{2}=4(pq+rs)^{2}-(p^{2}+q^{2}-r^{2}-s^{2})^{2},\,}](https://www.marefa.org/api/rest_v1/media/math/render/svg/060450a58b6bb8a5384b9245f3c522a4c7fb0036)
which is of the form
and hence can be written as
![{\displaystyle (2(pq+rs)-p^{2}-q^{2}+r^{2}+s^{2})(2(pq+rs)+p^{2}+q^{2}-r^{2}-s^{2})\,}](https://www.marefa.org/api/rest_v1/media/math/render/svg/673c1135722ba74f9233fb9762d22e4a11f94f93)
which, regrouping, is of the form
![{\displaystyle =((r+s)^{2}-(p-q)^{2})((p+q)^{2}-(r-s)^{2})\,}](https://www.marefa.org/api/rest_v1/media/math/render/svg/a153473c046ba9031a1342e9b7b9abd658e391ed)
hence yielding four linear factors:
![{\displaystyle =(q+r+s-p)(p+r+s-q)(p+q+s-r)(p+q+r-s).\,}](https://www.marefa.org/api/rest_v1/media/math/render/svg/9ecda588822917555cf3b80a618a57442d6b712e)
Introducing
![{\displaystyle 16({\mbox{Area}})^{2}=16(S-p)(S-q)(S-r)(S-s).\,}](https://www.marefa.org/api/rest_v1/media/math/render/svg/50dae6278c913069b67bb4b94f2d7e8662b25bbf)
Taking the square root, we get
![{\displaystyle {\mbox{Area}}={\sqrt {(S-p)(S-q)(S-r)(S-s)}}.}](https://www.marefa.org/api/rest_v1/media/math/render/svg/0b0f97b50db3d8033558a1411d5a62fccd5f2202)
انظر أيضاً
وصلات خارجية
Eric W. Weisstein, معادلة براهماگوپتا at MathWorld.