Fo rmula Gero na dozvolyaye viznachiti ploshu trikutnika S displaystyle S za danimi dovzhinami jogo storin a displaystyle a b displaystyle b i c displaystyle c Trikutnik iz storonami a b j c S p p a p b p c displaystyle S sqrt p p a p b p c de p a b c 2 displaystyle p frac a b c 2 polovina perimetru trikutnika abo pivperimetr Takozh rozpisuyuchi viraz pid korenem i vikoristovuyuchi formuli dlya kvadrata dvochlena i riznici kvadrativ mozhna oderzhati ekvivalentni varianti formuli S 1 4 a b c a b c a b c a b c displaystyle S frac 1 4 sqrt a b c a b c a b c a b c S 1 4 2 a 2 b 2 a 2 c 2 b 2 c 2 a 4 b 4 c 4 displaystyle S frac 1 4 sqrt 2 a 2 b 2 a 2 c 2 b 2 c 2 a 4 b 4 c 4 S 1 4 a 2 b 2 c 2 2 2 a 4 b 4 c 4 displaystyle S frac 1 4 sqrt a 2 b 2 c 2 2 2 a 4 b 4 c 4 S 1 4 4 a 2 b 2 a 2 c 2 b 2 c 2 a 2 b 2 c 2 2 displaystyle S frac 1 4 sqrt 4 a 2 b 2 a 2 c 2 b 2 c 2 a 2 b 2 c 2 2 S 1 4 4 a 2 b 2 a 2 b 2 c 2 2 displaystyle S frac 1 4 sqrt 4a 2 b 2 a 2 b 2 c 2 2 Zmist 1 Dovedennya trigonometrichne 2 Dovedennya geometrichne 3 Variaciyi j uzagalnennya 3 1 Formula Gerona Tartalyi 3 2 Teorema Lyuyilye 3 3 Formula Bramagupti 4 Primitki 5 PosilannyaDovedennya trigonometrichne RedaguvatiVizmemo shiroko vidomu formulu obchislennya ploshi trikutnika S 1 2 a b sin g displaystyle S 1 over 2 ab cdot sin gamma nbsp de g displaystyle gamma nbsp kut trikutnika sho lezhit navproti storoni c displaystyle c nbsp Zgidno z teoremoyu kosinusiv c 2 a 2 b 2 2 a b cos g displaystyle c 2 a 2 b 2 2ab cdot cos gamma nbsp Zvidsi cos g a 2 b 2 c 2 2 a b displaystyle cos gamma a 2 b 2 c 2 over 2ab nbsp Tomu sin 2 g 1 cos 2 g 1 cos g 1 cos g displaystyle sin 2 gamma 1 cos 2 gamma 1 cos gamma 1 cos gamma nbsp 2 a b a 2 b 2 c 2 2 a b 2 a b a 2 b 2 c 2 2 a b c 2 a b 2 2 a b a b 2 c 2 2 a b displaystyle 2ab a 2 b 2 c 2 over 2ab cdot 2ab a 2 b 2 c 2 over 2ab c 2 a b 2 over 2ab cdot a b 2 c 2 over 2ab nbsp 1 4 a 2 b 2 c a b c a b a b c a b c displaystyle 1 over 4a 2 b 2 c a b c a b a b c a b c nbsp Oskilki spravedlivi rivnosti a b c 2 p displaystyle a b c 2p nbsp a b c 2 p 2 c displaystyle a b c 2p 2c nbsp a c b 2 p 2 b displaystyle a c b 2p 2b nbsp c a b 2 p 2 a displaystyle c a b 2p 2a nbsp otrimuyemo sho sin g 2 a b p p a p b p c displaystyle sin gamma 2 over ab sqrt p p a p b p c nbsp Takim chinom S 1 2 a b sin g p p a p b p c displaystyle S 1 over 2 ab sin gamma sqrt p p a p b p c nbsp Dovedennya geometrichne Redaguvati nbsp Ilyustraciya do dovedennya formuli Gerona za dopomogoyu zovnivpisanogo kolaNehaj dano trikutnik A B C displaystyle ABC nbsp w 1 displaystyle w 1 nbsp ta w 2 displaystyle w 2 nbsp vpisane ta zovnivpisane yake dotikayetsya do storoni B C displaystyle BC nbsp kolo vidpovidno I displaystyle I nbsp centr vpisanogo kola w 1 displaystyle w 1 nbsp incentr tochka peretinu bisektris I a displaystyle I a nbsp centr zovnivpisanogo kola w 2 displaystyle w 2 nbsp tochka peretinu vnutrishnoyi ta dvoh zovnishnih bisektris Nehaj K displaystyle K nbsp tochka dotiku vpisanogo kola do storoni A B displaystyle AB nbsp a T displaystyle T nbsp tochka dotiku zovnivpisanogo kola do prodovzhennya storoni A B displaystyle AB nbsp Todi I K r displaystyle IK r nbsp radius vpisanogo kola w 1 displaystyle w 1 nbsp I a T r a displaystyle I a T r a nbsp radius zovnivpisanogo kola w 2 displaystyle w 2 nbsp i nehaj p a b c 2 displaystyle p frac a b c 2 nbsp pivperimetr trikutnika A B C displaystyle ABC nbsp Z vlastivostej vpisanogo ta zovnivpisanih kil vidomo sho A K p a displaystyle AK p a nbsp K B p b displaystyle KB p b nbsp B T p c displaystyle BT p c nbsp a A T p displaystyle AT p nbsp prichomu I K A B displaystyle IK perp AB nbsp ta I a T A B displaystyle I a T perp AB nbsp Zvidsi mayemo sho trikutniki A I K displaystyle AIK nbsp ta A I a T displaystyle AI a T nbsp podibni yak pryamokutni trikutniki zi spilnim gostrim kutom I A K displaystyle angle IAK nbsp Tomu A K A T I K I a T displaystyle frac AK AT frac IK I a T nbsp tobto p a p r r a displaystyle frac p a p frac r r a nbsp Zvidsi r a p a p r S displaystyle r a p a pr S nbsp Znajdemo kut B I a T displaystyle angle BI a T nbsp Oskilki B I a T displaystyle bigtriangleup BI a T nbsp pryamokutnij to B I a T 90 I a B T displaystyle angle BI a T 90 circ angle I a BT nbsp Za pobudovoyu B I a displaystyle BI a nbsp bisektrisa kuta C B T 180 B displaystyle angle CBT 180 circ angle B nbsp yak zovnishnij kut a tomu I a B T C B T 2 180 B 2 90 B 2 displaystyle angle I a BT frac angle CBT 2 frac 180 circ angle B 2 90 circ frac angle B 2 nbsp Zvidsi B I a T B 2 displaystyle angle BI a T frac angle B 2 nbsp Ale takozh B I K B 2 displaystyle angle BIK frac angle B 2 nbsp oskilki B I displaystyle BI nbsp bisektrisa kuta B displaystyle angle B nbsp Otrimali sho trikutniki B I a T displaystyle BI a T nbsp ta B I K displaystyle BIK nbsp podibni yak pryamokutni za rivnimi gostrimi kutami Tomu I a T B K B T I K displaystyle frac I a T BK frac BT IK nbsp tobto r a p b p c r displaystyle frac r a p b frac p c r nbsp Zvidsi r a r p b p c displaystyle r a r p b p c nbsp Z rivnostej r a p a p r S displaystyle r a p a pr S nbsp oderzhimo sho S 2 p p a r a r displaystyle S 2 p p a r a r nbsp Zaminivshi r a r displaystyle r a r nbsp po vishe dovedenij formuli r a r p b p c displaystyle r a r p b p c nbsp oderzhimo ostatochno S 2 p p a p b p c displaystyle S 2 p p a p b p c nbsp abo sho te same S p p a p b p c displaystyle S sqrt p p a p b p c nbsp Variaciyi j uzagalnennya RedaguvatiFormulu Gerona mozhna zapisati za dopomogoyu viznachnika u viglyadi 1 16 S 2 0 a 2 b 2 1 a 2 0 c 2 1 b 2 c 2 0 1 1 1 1 0 a b c 0 b a 0 c c 0 a b 0 c b a displaystyle 16S 2 begin vmatrix 0 amp a 2 amp b 2 amp 1 a 2 amp 0 amp c 2 amp 1 b 2 amp c 2 amp 0 amp 1 1 amp 1 amp 1 amp 0 end vmatrix begin vmatrix a amp b amp c amp 0 b amp a amp 0 amp c c amp 0 amp a amp b 0 amp c amp b amp a end vmatrix nbsp Pershij viznachnik ostannoyi formuli ye okremim vipadkom viznachnika Keli Mengera en dlya obchislennya giperob yemu simpleksa Nizka formul dlya ploshi trikutnika podibni za strukturoyu do formuli Gerona ale virazhayetsya cherez inshi parametri trikutnika Napriklad cherez dovzhini median m a displaystyle m a nbsp m b displaystyle m b nbsp i m c displaystyle m c nbsp i yih pivsumu s m a m b m c 2 displaystyle sigma m a m b m c 2 nbsp 2 S 4 3 s s m a s m b s m c displaystyle S frac 4 3 sqrt sigma sigma m a sigma m b sigma m c nbsp cherez dovzhini visot h a displaystyle h a nbsp h b displaystyle h b nbsp i h c displaystyle h c nbsp i pivsumu yih obernenih velichin H h a 1 h b 1 h c 1 2 displaystyle H h a 1 h b 1 h c 1 2 nbsp 3 S 1 4 H H h a 1 H h b 1 H h c 1 displaystyle S 1 4 sqrt H H h a 1 H h b 1 H h c 1 nbsp dd cherez kut trikutnika a displaystyle alpha nbsp b displaystyle beta nbsp i g displaystyle gamma nbsp pivsumu yih sinusiv s sin a sin b sin g 2 displaystyle s sin alpha sin beta sin gamma 2 nbsp i diametr opisanogo kola D a sin a b sin b c sin g displaystyle D tfrac a sin alpha tfrac b sin beta tfrac c sin gamma nbsp 4 S D 2 s s sin a s sin b s sin g displaystyle S D 2 sqrt s s sin alpha s sin beta s sin gamma nbsp dd Formula Gerona Tartalyi Redaguvati Dlya tetraedriv isnuye formula Gerona Tartalyi uzagalnena takozh na vipadok inshih bagatogrannikiv zginanij mnogogrannik yaksho v tetraedra dovzhini reber rivni l 1 l 2 l 3 l 4 l 5 l 6 displaystyle l 1 l 2 l 3 l 4 l 5 l 6 nbsp to dlya jogo ob yemu V displaystyle V nbsp istinnij viraz 144 V 2 l 1 2 l 5 2 l 2 2 l 3 2 l 4 2 l 6 2 l 1 2 l 5 2 l 2 2 l 6 2 l 1 2 l 3 2 l 4 2 l 5 2 l 2 2 l 6 2 l 3 2 l 4 2 l 1 2 l 2 2 l 5 2 l 6 2 l 3 2 l 4 2 l 1 2 l 2 2 l 4 2 l 2 2 l 3 2 l 5 2 l 1 2 l 3 2 l 6 2 l 4 2 l 5 2 l 6 2 displaystyle begin aligned 144V 2 amp l 1 2 l 5 2 l 2 2 l 3 2 l 4 2 l 6 2 l 1 2 l 5 2 amp l 2 2 l 6 2 l 1 2 l 3 2 l 4 2 l 5 2 l 2 2 l 6 2 amp l 3 2 l 4 2 l 1 2 l 2 2 l 5 2 l 6 2 l 3 2 l 4 2 amp l 1 2 l 2 2 l 4 2 l 2 2 l 3 2 l 5 2 l 1 2 l 3 2 l 6 2 l 4 2 l 5 2 l 6 2 end aligned nbsp Formulu Gerona Tartalyi mozhna vipisati dlya tetraedra v yavnomu viglyadi yaksho U displaystyle U nbsp V displaystyle V nbsp W displaystyle W nbsp u displaystyle u nbsp v displaystyle v nbsp w displaystyle w nbsp dovzhini reber tetraedra pershi tri z nih utvoryuyut trikutnik i napriklad rebro u displaystyle u nbsp protlezhne rebru U displaystyle U nbsp i tak dali to spravedlivi formuli 5 6 V a b c d a b c d a b c d a b c d 192 u v w displaystyle text V frac sqrt a b c d a b c d a b c d a b c d 192 u v w nbsp de a x Y Z b y Z X c z X Y d x y z X w U v U v w x U v w v w U Y u V w V w u y V w u w u V Z v W u W u v z W u v u v W displaystyle begin aligned a amp sqrt xYZ b amp sqrt yZX c amp sqrt zXY d amp sqrt xyz X amp w U v U v w x amp U v w v w U Y amp u V w V w u y amp V w u w u V Z amp v W u W u v z amp W u v u v W end aligned nbsp dd Teorema Lyuyilye Redaguvati Za teoremoyu Lyuyilye plosha sferichnogo trikutnika virazhayetsya cherez jogo storoni 8 a a R 8 b b R 8 c c R displaystyle theta a frac a R theta b frac b R theta c frac c R nbsp kak S 4 R 2 arctg tg 8 s 2 tg 8 s 8 a 2 tg 8 s 8 b 2 tg 8 s 8 c 2 displaystyle S 4R 2 operatorname arctg sqrt operatorname tg left frac theta s 2 right operatorname tg left frac theta s theta a 2 right operatorname tg left frac theta s theta b 2 right operatorname tg left frac theta s theta c 2 right nbsp de 8 s 8 a 8 b 8 c 2 displaystyle theta s frac theta a theta b theta c 2 nbsp pivperimetr Formula Bramagupti Redaguvati Dokladnishe Formula BramaguptiFormula Bramagupti ye uzagalnennyam formuli Gerona dlya ploshi trikutnika A same plosha S vpisanogo u kolo chotirikutnika zi storonami a b c d i pivperimetr p dorivnyuye S p a p b p c p d displaystyle S sqrt p a p b p c p d nbsp U comu vipadku trikutnik viyavlyayetsya granichnim vipadkom upisanogo chotirikutnika pri pryamuvanni dovzhini odniyeyi zi storin do nulya Ta zh formula Brahmagupti cherez viznachnik 7 S 1 4 a b c d b a d c c d a b d c b a displaystyle S frac 1 4 sqrt 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 Primitki Redaguvati Weisstein Eric W Heron s Formula Arhivovano 5 veresnya 2015 u Wayback Machine From MathWorld A Wolfram Web Resource Benyi Arpad A Heron type formula for the triangle Mathematical Gazette 87 July 2003 324 326 Mitchell Douglas W A Heron type formula for the reciprocal area of a triangle Mathematical Gazette 89 November 2005 494 Mitchell Douglas W A Heron type area formula in terms of sines Mathematical Gazette 93 March 2009 108 109 W Kahan What has the Volume of a Tetrahedron to do with Computer Programming Languages 1 Arhivovano 27 chervnya 2013 u Wayback Machine pp 16 17 Markelov S Formula dlya obyoma tetraedra Matematicheskoe prosveshenie Vyp 6 2002 S 132 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 39Posilannya RedaguvatiWeisstein Eric W Formula Gerona angl na sajti Wolfram MathWorld Kushnir I A Geometriya Poisk i vdohnovenie M MCNMO 2013 592 s il ISBN 978 5 4439 0058 2 nbsp Ce nezavershena stattya z geometriyi Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Otrimano z https uk wikipedia org w index php title Formula Gerona amp oldid 38117280
