Fo rmula Bramagu pti angl Brahmagupta s formula matematichna formula yaka virazhaye ploshu vpisanogo u kolo chotirikutnika yak funkciyu dovzhin jogo storin Yaksho vpisanij u kolo chotirikutnik maye dovzhini storin a b c d displaystyle a b c d i pivperimetr p a b c d 2 displaystyle p frac a b c d 2 to jogo plosha S displaystyle S virazhayetsya formuloyu S p a p b p c p d displaystyle S sqrt p a p b p c p d Dovedennya Plosha vpisanogo u kolo chotirikutnika A B C D displaystyle ABCD zi storonami a b c d dorivnyuye sumi plosh A C B displaystyle triangle ACB ta B D C displaystyle triangle BDC S 1 2 a b sin A 1 2 c d sin C displaystyle S frac 1 2 ab sin A frac 1 2 cd sin C Tak yak A B C D displaystyle ABCD ye vpisanim chotirikutnikom to D A B 180 D C B displaystyle angle DAB 180 circ angle DCB Otzhe sin A sin C displaystyle sin A sin C S 1 2 a b sin A 1 2 c d sin A displaystyle S frac 1 2 ab sin A frac 1 2 cd sin A S 2 1 4 sin 2 A a b c d 2 displaystyle S 2 frac 1 4 sin 2 A ab cd 2 4 S 2 1 cos 2 A a b c d 2 displaystyle 4S 2 1 cos 2 A ab cd 2 4 S 2 a b c d 2 cos 2 A a b c d 2 displaystyle 4S 2 ab cd 2 cos 2 A ab cd 2 Zapisavshi teoremu kosinusiv dlya storoni C B displaystyle CB u A C B displaystyle triangle ACB i B D C displaystyle triangle BDC otrimuyemo a 2 b 2 2 a b cos A c 2 d 2 2 c d cos C displaystyle a 2 b 2 2ab cos A c 2 d 2 2cd cos C Skoristayemos zalezhnistyu cos C cos A displaystyle cos C cos A A displaystyle A i C displaystyle C protilezhni kuti pislya chogo vinesemo za duzhki 2 cos A displaystyle 2 cos A 2 cos A a b c d a 2 b 2 c 2 d 2 displaystyle 2 cos A ab cd a 2 b 2 c 2 d 2 Pidstavimo otrimane u zapisanu vishe formulu ploshi 4 S 2 a b c d 2 1 4 a 2 b 2 c 2 d 2 2 displaystyle 4S 2 ab cd 2 frac 1 4 a 2 b 2 c 2 d 2 2 16 S 2 4 a b c d 2 a 2 b 2 c 2 d 2 2 displaystyle 16S 2 4 ab cd 2 a 2 b 2 c 2 d 2 2 Zastosuyemo formulu x 2 y 2 x y x y displaystyle x 2 y 2 x y x y 2 a b c d a 2 b 2 c 2 d 2 2 a b c d a 2 b 2 c 2 d 2 displaystyle 2 ab cd a 2 b 2 c 2 d 2 2 ab cd a 2 b 2 c 2 d 2 a b 2 c d 2 c d 2 a b 2 displaystyle a b 2 c d 2 c d 2 a b 2 a b c d a b d c a c d b b c d a displaystyle a b c d a b d c a c d b b c d a Tak yak pivperimetr p a b c d 2 displaystyle p frac a b c d 2 16 S 2 16 p a p b p c p d displaystyle 16S 2 16 p a p b p c p d Dobuvayuchi kvadratnij korin otrimuyemo S p a p b p c p d displaystyle S sqrt p a p b p c p d Zmist 1 Variaciyi j uzagalnennya 2 Div takozh 3 Primitki 4 DzherelaVariaciyi j uzagalnennya RedaguvatiFormula Bramagupti uzagalnyuye formulu Gerona dlya viznachennya ploshi trikutnika na vipadok vpisanogo u kolo chotirikutnika dostatno vvazhati sho dovzhina odniyeyi iz storin dorivnyuye nulyu napriklad d 0 displaystyle d 0 nbsp i formula Bramagupti zvoditsya do formuli Gerona Na vipadok dovilnih chotirikutnikiv formula Bramagupti mozhe buti uzagalnena tak S p a p b p c p d a b c d cos 2 8 displaystyle S sqrt p a p b p c p d abcd cos 2 theta nbsp dd de 8 displaystyle theta nbsp pivsuma protilezhnih kutiv chotirikutnika Yaku same paru protilezhnih kutiv vzyati roli ne vidigraye tak yak yaksho pivsuma odniyeyi pari protilezhnih kutiv dorivnyuye 8 displaystyle theta nbsp to pivsuma dvoh inshih kutiv bude 180 8 displaystyle 180 circ theta nbsp i cos 2 180 8 cos 2 8 displaystyle cos 2 180 circ theta cos 2 theta nbsp Inkoli cyu zagalnishu formulu zapisuyut tak S p a p b p c p d 1 4 a c b d u v a c b d u v displaystyle S sqrt p a p b p c p d textstyle 1 over 4 ac bd uv ac bd uv nbsp dd de u displaystyle u nbsp i v displaystyle v nbsp dovzhini diagonalej chotirikutnika Matematik Devid P Robbins angl David P Robbins doviv 1 sho dlya dovilnogo vpisanogo mnogokutnika z n displaystyle n nbsp storonami velichina 4 S 2 displaystyle 4S 2 nbsp ye korenem deyakogo mnogochlena P displaystyle P nbsp koeficiyenti yakogo u svoyu chergu ye mnogochlenami vid dovzhin storin Vin znajshov ci mnogochleni dlya n 5 displaystyle n 5 nbsp ta n 6 displaystyle n 6 nbsp Inshimi avtorami vstanovleno 2 sho mnogochlen P displaystyle P nbsp mozhna obrati tak shob jogo starshij koeficiyent dorivnyuvav odinici a stepin N N n displaystyle N N n nbsp dorivnyuvav D k displaystyle Delta k nbsp pri n 2 k 1 displaystyle n 2k 1 nbsp i 2 D k displaystyle 2 Delta k nbsp yaksho n 2 k 2 displaystyle n 2k 2 nbsp TutD k 2 k 1 2 2 k k 2 2 k 1 j 0 k 1 k j 2 k 1 j displaystyle Delta k frac 2k 1 2 binom 2k k 2 2k 1 sum j 0 k 1 k j binom 2k 1 j nbsp dd de k j k j k j displaystyle tbinom k j tfrac k j k j nbsp binomialni koeficiyenti Dlya mnogokutnikiv z nevelikim chislom storin mayemo D 1 1 displaystyle Delta 1 1 nbsp D 2 7 displaystyle Delta 2 7 nbsp D 3 38 displaystyle Delta 3 38 nbsp D 4 187 displaystyle Delta 4 187 dots nbsp poslidovnist A000531 z Onlajn enciklopediyi poslidovnostej cilih chisel OEIS i N 4 2 displaystyle N 4 2 nbsp N 5 7 displaystyle N 5 7 nbsp N 6 14 displaystyle N 6 14 nbsp N 7 38 displaystyle N 7 38 dots nbsp poslidovnist A107373 z Onlajn enciklopediyi poslidovnostej cilih chisel OEIS Yaksho u formuli Bramagupti viraziti pivperimetr cherez pivsumu usih storin danogo chotirikutnika pidnesti obidvi chastini do kvadratu pomnozhiti na 16 rozkriti duzhki ta zvesti podibni to vona nabude viglyadu 16 S 2 a 4 b 4 c 4 d 4 2 a 2 b 2 b 2 c 2 a 2 c 2 a 2 d 2 b 2 d 2 c 2 d 2 8 a b c d displaystyle 16S 2 a 4 b 4 c 4 d 4 2 a 2 b 2 b 2 c 2 a 2 c 2 a 2 d 2 b 2 d 2 c 2 d 2 8abcd nbsp dd Prava chastina rivnyannya bude zbigatis z rozkladom viznachnika podanogo nizhche yaksho jogo pomnozhiti na 1 Tomu mozhna napisati sho 3 16 S 2 a b c d b a d c c d a b d c b a displaystyle 16S 2 begin vmatrix a amp b amp c amp d b amp a amp d amp c c amp d amp a amp b d amp c amp b amp a end vmatrix nbsp dd Div takozh RedaguvatiBramagupta Teorema BramaguptiPrimitki Redaguvati D P Robbins Areas of polygons inscribed in a circle Discrete amp Computational Geometry 12 1994 P 223 236 Maley F Miller Robbins David P Roskies Julie 2005 On the areas of cyclic and semicyclic polygons Advances in Applied Mathematics 34 4 669 689 doi 10 1016 j aam 2004 09 008 Arhiv originalu za 10 sichnya 2020 Procitovano 3 serpnya 2016 Starikov V N Zametki po geometrii Nauchnyj poisk gumanitarnye i socialno ekonomicheskie nauki sbornik nauchnyh trudov Vypusk 1 Gl red Romanova I V Cheboksary CDIP INet 2014 S 37 39Dzherela RedaguvatiPrasolov V V Formula Brahmagupty Matematika v shkole 1991 5 G M Kokseter S Grejtcer Novye vstrechi s geometriej Pod red A P Savina M Nauka 1978 S 73 74 Biblioteka matematicheskogo kruzhka Vyp 14 Varfolomeev V V Vpisannye mnogougolniki i polinomy Gerona Matematicheskij sbornik 2003 T 194 3 S 3 24 M Fedorchuk I Pak 2005 Rigidity and polynomial invariants of convex polytopes Duke Mathematical Journal 129 2 371 404 doi 10 1215 S0012 7094 05 12926 X Arhiv originalu za 3 zhovtnya 2013 Procitovano 3 serpnya 2016 Otrimano z https uk wikipedia org w index php title Formula Bramagupti amp oldid 35117248
