Definición Dado un triángulo de referencia cuyos lados son a , b y c y cuyos ángulos internos correspondientes son A , B y C , la notación del triángulo de Conway se representa simplemente de la siguiente manera: S = b do pecado A = a do pecado B = a b pecado do , {\displaystyle S=bc\sin A=ac\sin B=ab\sin C,} [ 3 ] [ 4 ] S = 2 × área del triángulo de referencia , S φ = S cuna φ . {\displaystyle {\begin{aligned}S&=2\times {\text{área del triángulo de referencia}},\\[2pt]S_{\varphi }&=S\cot \varphi .\end{aligned}}}
En particular: S A = S cuna A = b do porque A = b 2 + do 2 − a 2 2 , S B = S cuna B = a do porque B = a 2 + do 2 − b 2 2 , S do = S cuna do = a b porque do = a 2 + b 2 − do 2 2 , S ω = S cuna ω = a 2 + b 2 + do 2 2 , {\displaystyle {\begin{aligned}S_{A}&=S\cot A=bc\cos A={\frac {b^{2}+c^{2}-a^{2}}{2}},\\[2pt]S_{B}&=S\cot B=ac\cos B={\frac {a^{2}+c^{2}-b^{2}}{2}},\\[2pt]S_{C}&=S\cot C=ab\cos C={\frac {a^{2}+b^{2}-c^{2}}{2}},\\[2pt]S_{\omega }&=S\cot \omega ={\frac {a^{2}+b^{2}+c^{2}}{2}},\end{aligned}}}
donde ω ángulo de Brocard . Se utiliza la ley de los cosenos : a 2 = b 2 + do 2 − 2 b do porque A . {\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos A.}
Identidades de ángulos de tercio, doble y medio: S π 3 = S cuna π 3 = S 3 3 S 2 φ = S φ 2 − S 2 2 S φ S φ 2 = S φ + S φ 2 + S 2 {\displaystyle {\begin{aligned}S_{\frac {\pi }{3}}&=S\cot {\frac {\pi }{3}}=S{\frac {\sqrt {3}}{3}}\\[2pt]S_{2\varphi }&={\frac {S_{\varphi }^{2}-S^{2}}{2S_{\varphi }}}\\[2pt]S_{\frac {\varphi }{2}}&=S_{\varphi }+{\sqrt {S_{\varphi }^{2}+S^{2}}}\end{aligned}}}
para valores de φ 0 < φ π ,
S ϑ + φ = S ϑ S φ − S 2 S ϑ + S φ , S ϑ − φ = S ϑ S φ + S 2 S φ − S ϑ . {\displaystyle {\begin{aligned}S_{\vartheta +\varphi }&={\frac {S_{\vartheta }S_{\varphi }-S^{2}}{S_{\vartheta }+S_{\varphi }}},\\[2pt]S_{\vartheta -\varphi }&={\frac {S_{\vartheta }S_{\varphi }+S^{2}}{S_{\varphi}-S_{\vartheta }}}.\end{aligned}}}
Además, la convención utiliza una notación abreviada paraS ϑ S φ = S ϑ φ , {\displaystyle S_{\vartheta }S_{\varphi }=S_{\vartheta \varphi },} S ϑ S φ S ψ = S ϑ φ ψ . {\displaystyle S_{\vartheta }S_{\varphi }S_{\psi }=S_{\vartheta \varphi \psi }.}
Relaciones trigonométricas pecado A = S b do = S S A 2 + S 2 porque A = S A b do = S A S A 2 + S 2 broncearse A = S S A a 2 = S B + S do b 2 = S A + S do do 2 = S A + S B {\displaystyle {\begin{aligned}\sin A&={\frac {S}{bc}}={\frac {S}{\sqrt {S_{A}^{2}+S^{2}}}}\\[2pt]\cos A&={\frac {S_{A}}{bc}}={\frac {S_{A}}{\sqrt {S_{A}^{2}+S^{2}}}}\\[2pt]\tan A&={\frac {S}{S_{A}}}\\[6pt]a^{2}&=S_{B}+S_{C}\\[2pt]b^{2}&=S_{A}+S_{C}\\[2pt]c^{2}&=S_{A}+S_{B}\end{aligned}}}
Identidades importantes ∑ cíclico S A = S A + S B + S do = S ω S 2 = b 2 do 2 − S A 2 = a 2 do 2 − S B 2 = a 2 b 2 − S do 2 S B do = S B S do = S 2 − a 2 S A S A do = S A S do = S 2 − b 2 S B S A B = S A S B = S 2 − do 2 S do S A B do = S A S B S do = S 2 ( S ω − 4 R 2 ) S ω = s 2 − r 2 − 4 r R {\displaystyle {\begin{aligned}\sum _{\text{cyclic}}S_{A}&=S_{A}+S_{B}+S_{C}=S_{\omega }\\[2pt]S^{2}&=b^{2}c^{2}-S_{A}^{2}=a^{2}c^{2}-S_{B}^{2}=a^{2}b^{2}-S_{C}^{2}\\[6pt]S_{BC}&=S_{B}S_{C}=S^{2}-a^{2}S_{A}\\[2pt]S_{AC}&=S_{A}S_{C}=S^{2}-b^{2}S_{B}\\[2pt]S_{AB}&=S_{A}S_{B}=S^{2}-c^{2}S_{C}\\[8pt]S_{ABC}&=S_{A}S_{B}S_{C}=S^{2}(S_{\omega }-4R^{2})\\[8pt]S_{\omega }&=s^{2}-r^{2}-4rR\end{aligned}}}
dónde
R es el radio de la circunferencia circunscrita. r es el incentro a b do = 2 S R {\displaystyle abc=2SR} s = a + b + do 2 {\displaystyle s={\frac {a+b+c}{2}}} S r = a + b + do {\displaystyle {\frac {S}{r}}=a+b+c}
Conversiones trigonométricas pecado A pecado B pecado do = S 4 R 2 porque A porque B porque do = S ω − 4 R 2 4 R 2 {\displaystyle {\begin{aligned}\sin A\sin B\sin C&={\frac {S}{4R^{2}}}\\[2pt]\cos A\cos B\cos C&={\frac {S_{\omega }-4R^{2}}{4R^{2}}}\end{aligned}}}
∑ cíclico pecado A = S 2 R r = s R ∑ cíclico porque A = r + R R ∑ cíclico broncearse A = S S ω − 4 R 2 = broncearse A broncearse B broncearse do {\displaystyle {\begin{aligned}\sum _{\text{cyclic}}\sin A&={\frac {S}{2Rr}}={\frac {s}{R}}\\[2pt]\sum _{\text{cyclic}}\cos A&={\frac {r+R}{R}}\\[2pt]\sum _{\text{cyclic}}\tan A&={\frac {S}{S_{\omega }-4R^{2}}}=\tan A\tan B\tan C\end{aligned}}}
∑ cíclico a 2 S A = a 2 S A + b 2 S B + do 2 S do = 2 S 2 ∑ cíclico a 4 = 2 ( S ω 2 − S 2 ) ∑ cíclico S A 2 = S ω 2 − 2 S 2 ∑ cíclico S B do = ∑ cíclico S B S do = S 2 ∑ cíclico b 2 do 2 = S ω 2 + S 2 {\displaystyle {\begin{array}{ll}\displaystyle \sum _{\text{cyclic}}a^{2}S_{A}\!\!&=&a^{2}S_{A}+b^{2}S_{B}+c^{2}S_{C}=2S^{2}\\[2pt]\displaystyle \sum _{\text{cyclic}}a^{4}&=&2(S_{\omega }^{2}-S^{2})\\[2pt]\displaystyle \sum _{\text{cyclic}}S_{A}^{2}&=&S_{\omega }^{2}-2S^{2}\\[2pt]\displaystyle \sum _{\text{cyclic}}S_{BC}&=&\displaystyle \sum _{\text{cyclic}}S_{B}S_{C}=S^{2}\\[2pt]\displaystyle \sum _{\text{cyclic}}b^{2}c^{2}&=&S_{\omega }^{2}+S^{2}\end{array}}}
Aplicaciones Sea D la distancia entre dos puntos P y Q cuyas coordenadas trilineales son PAG = pag a : pag b : pag do , Q = q a : q b : q do . {\displaystyle {\begin{aligned}P=p_{a}:p_{b}:p_{c},\\Q=q_{a}:q_{b}:q_{c}.\end{aligned}}} K pag = a pag a + b pag b + do pag do , K q = a q a + b q b + do q do . {\displaystyle {\begin{aligned}K_{p}&=ap_{a}+bp_{b}+cp_{c},\\K_{q}&=aq_{a}+bq_{b}+cq_{c}.\end{aligned}}} D viene dado por la fórmula: [ 5 ] D 2 = ∑ cíclico a 2 S A ( pag a K pag − q a K q ) 2 {\displaystyle D^{2}=\sum _{\text{cyclic}}a^{2}S_{A}\left({\frac {p_{a}}{K_{p}}}-{\frac {q_{a}}{K_{q}}}\right)^{2}}
Distancia entre el circuncentro y el ortocentro Utilizando esta fórmula es posible determinar | OH | , la distancia entre el circuncentro y el ortocentro, de la siguiente manera:
Para el circuncentropag a = a S A , {\displaystyle p_{a}=aS_{A},}
y para el ortocentroq a = S B S do a , {\displaystyle q_{a}={\frac {S_{B}S_{C}}{a}},}
K pag = ∑ cíclico a 2 S A = 2 S 2 , K q = ∑ cíclico S B S do = S 2 . {\displaystyle {\begin{aligned}K_{p}&=\sum _{\text{cyclic}}a^{2}S_{A}=2S^{2},\\[2pt]K_{q}&=\sum _{\text{cyclic}}S_{B}S_{C}=S^{2}.\end{aligned}}}
Por eso:
D 2 = ∑ cíclico a 2 S A ( a S A 2 S 2 − S B S do a S 2 ) 2 = 1 4 S 4 ∑ cíclico a 4 S A 3 − S A S B S do S 4 ∑ cíclico a 2 S A + S A S B S do S 4 ∑ cíclico S B S do = 1 4 S 4 ∑ cíclico a 2 S A 2 ( S 2 − S B S do ) − 2 ( S ω − 4 R 2 ) + ( S ω − 4 R 2 ) = 1 4 S 2 ∑ cíclico a 2 S A 2 − S A S B S do S 4 ∑ cíclico a 2 S A − ( S ω − 4 R 2 ) = 1 4 S 2 ∑ cíclico a 2 ( b 2 do 2 − S 2 ) − 1 2 ( S ω − 4 R 2 ) − ( S ω − 4 R 2 ) = 3 a 2 b 2 do 2 4 S 2 − 1 4 ∑ cíclico a 2 − 3 2 ( S ω − 4 R 2 ) = 3 R 2 − 1 2 S ω − 3 2 S ω + 6 R 2 = 9 R 2 − 2 S ω . {\displaystyle {\begin{aligned}D^{2}&=\sum _{\text{cyclic}}a^{2}S_{A}\left({\frac {aS_{A}}{2S^{2}}}-{\frac {S_{B}S_{C}}{aS^{2}}}\right)^{2}\\[2pt]&={\frac {1}{4S^{4}}}\sum _{\text{cyclic}}a^{4}S_{A}^{3}-{\frac {S_{A}S_{B}S_{C}}{S^{4}}}\sum _{\text{cyclic}}a^{2}S_{A}+{\frac {S_{A}S_{B}S_{C}}{S^{4}}}\sum _{\text{cyclic}}S_{B}S_{C}\\[2pt]&={\frac {1}{4S^{4}}}\sum _{\text{cyclic}}a^{2}S_{A}^{2}(S^{2}-S_{B}S_{C})-2(S_{\omega }-4R^{2})+(S_{\omega }-4R^{2})\\[2pt]&={\frac {1}{4S^{2}}}\sum _{\text{cyclic}}a^{2}S_{A}^{2}-{\frac {S_{A}S_{B}S_{C}}{S^{4}}}\sum _{\text{cyclic}}a^{2}S_{A}-(S_{\omega }-4R^{2})\\[2pt]&={\frac {1}{4S^{2}}}\sum _{\text{cyclic}}a^{2}(b^{2}c^{2}-S^{2})-{\frac {1}{2}}(S_{\omega }-4R^{2})-(S_{\omega }-4R^{2})\\[2pt]&={\frac {3a^{2}b^{2}c^{2}}{4S^{2}}}-{\frac {1}{4}}\sum _{\text{cyclic}}a^{2}-{\frac {3}{2}}(S_{\omega }-4R^{2})\\[2pt]&=3R^{2}-{\frac {1}{2}}S_{\omega }-{\frac {3}{2}}S_{\omega }+6R^{2}\\[10pt]&=9R^{2}-2S_{\omega }.\end{aligned}}}
Por lo tanto, [ 6 ]
| O H | = 9 R 2 − 2 S ω . {\displaystyle |OH|={\sqrt {9R^{2}-2S_{\omega }}}.}
Referencias ↑ Chen, Evan (2016). Geometría euclidiana en las Olimpiadas Matemáticas . Asociación Matemática de América . pág. 132. ISBN 978-0883858394 . ↑ Loriga, Juan Jacobo Durán, "Nota sobre el triángulo", en El Progreso Matemático, tomo IV (1894), páginas 313-316. , Periódico de Matemáticas Puras y Aplicadas ↑ Yiu, Paul (2002), "Notación." §3.4.1 en Introducción a la geometría del triángulo. pp. 33-34, Versión 2.0402, abril de 2002 (PDF) , Departamento de Matemáticas , Universidad Atlántica de Florida, pp. 33–34 ↑ Kimberling, Clark, Enciclopedia de Centros Triangulares - ETC, Parte 1 "Introducido el 1 de noviembre de 2011: Combinaciones" Nota 6 , Universidad de Evansville ↑ Yiu, Paul (2002), "La fórmula de la distancia" §7.1 en Introducción a la geometría del triángulo. pág. 87, Versión 2.0402, abril de 2002 (PDF) , Departamento de Matemáticas, Universidad Atlántica de Florida, pág. 87 ↑ Weisstein, Eric W. "Ortocentro §(14)" . MathWorld . 
Hispanopedia Wiki
![{\displaystyle {\begin{aligned}S&=2\times {\text{área del triángulo de referencia}},\\[2pt]S_{\varphi }&=S\cot \varphi .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cb1c2014aed51eb1571fbb1dfd1d750b1f1050)
![{\displaystyle {\begin{aligned}S_{A}&=S\cot A=bc\cos A={\frac {b^{2}+c^{2}-a^{2}}{2}},\\[2pt]S_{B}&=S\cot B=ac\cos B={\frac {a^{2}+c^{2}-b^{2}}{2}},\\[2pt]S_{C}&=S\cot C=ab\cos C={\frac {a^{2}+b^{2}-c^{2}}{2}},\\[2pt]S_{\omega }&=S\cot \omega ={\frac {a^{2}+b^{2}+c^{2}}{2}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f8be09d8bf8f3ae2da9b3ef6dc3760aae07fd6)
![{\displaystyle {\begin{aligned}S_{\frac {\pi }{3}}&=S\cot {\frac {\pi }{3}}=S{\frac {\sqrt {3}}{3}}\\[2pt]S_{2\varphi }&={\frac {S_{\varphi }^{2}-S^{2}}{2S_{\varphi }}}\\[2pt]S_{\frac {\varphi }{2}}&=S_{\varphi }+{\sqrt {S_{\varphi }^{2}+S^{2}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02196a2df49d91533a038ff6b7173ab1ecda1eaa)
![{\displaystyle {\begin{aligned}S_{\vartheta +\varphi }&={\frac {S_{\vartheta }S_{\varphi }-S^{2}}{S_{\vartheta }+S_{\varphi }}},\\[2pt]S_{\vartheta -\varphi }&={\frac {S_{\vartheta }S_{\varphi }+S^{2}}{S_{\varphi}-S_{\vartheta }}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45666fbd0ab88d70f4e3828605395b77eb21fee2)
![{\displaystyle {\begin{aligned}\sin A&={\frac {S}{bc}}={\frac {S}{\sqrt {S_{A}^{2}+S^{2}}}}\\[2pt]\cos A&={\frac {S_{A}}{bc}}={\frac {S_{A}}{\sqrt {S_{A}^{2}+S^{2}}}}\\[2pt]\tan A&={\frac {S}{S_{A}}}\\[6pt]a^{2}&=S_{B}+S_{C}\\[2pt]b^{2}&=S_{A}+S_{C}\\[2pt]c^{2}&=S_{A}+S_{B}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d195822308964c8d0e692f6cf649784f47313b3)
![{\displaystyle {\begin{aligned}\sum _{\text{cíclico}}S_{A}&=S_{A}+S_{B}+S_{C}=S_{\omega }\\[2pt]S^{2}&=b^{2}c^{2}-S_{A}^{2}=a^{2}c^{2}-S_{B}^{2}=a^{2}b^{2}-S_{C}^{2}\\[6pt]S_{BC}&=S_{B}S_{C}=S^{2}-a^{2}S_{A}\\[2pt]S_{AC}&=S_{A}S_{C}=S^{2}-b^{2}S_{B}\\[2pt]S_{AB}&=S_{A}S_{B}=S^{2}-c^{2}S_{C}\\[8pt]S_{ABC}&=S_{A}S_{B}S_{C}=S^{2}(S_{\omega }-4R^{2})\\[8pt]S_{\omega }&=s^{2}-r^{2}-4rR\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/889a062df471a439dd7b932cc6b48638b703d589)
![{\displaystyle {\begin{aligned}\sin A\sin B\sin C&={\frac {S}{4R^{2}}}\\[2pt]\cos A\cos B\cos C&={\frac {S_{\omega }-4R^{2}}{4R^{2}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/999be973dc3adffa1d31b7400bb5ae82616920c1)
![{\displaystyle {\begin{aligned}\sum _{\text{cíclico}}\sin A&={\frac {S}{2Rr}}={\frac {s}{R}}\\[2pt]\sum _{\text{cíclico}}\cos A&={\frac {r+R}{R}}\\[2pt]\sum _{\text{cíclico}}\tan A&={\frac {S}{S_{\omega }-4R^{2}}}=\tan A\tan B\tan C\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2b8ebd566b2c3050f62b7c3a5499109dde406d)
![{\displaystyle {\begin{array}{ll}\displaystyle \sum _{\text{cíclico}}a^{2}S_{A}\!\!&=&a^{2}S_{A}+b^{2}S_{B}+c^{2}S_{C}=2S^{2}\\[2pt]\displaystyle \sum _{\text{cíclico}}a^{4}&=&2(S_{\omega }^{2}-S^{2})\\[2pt]\displaystyle \sum _{\text{cíclico}}S_{A}^{2}&=&S_{\omega }^{2}-2S^{2}\\[2pt]\displaystyle \sum _{\text{cíclico}}S_{BC}&=&\displaystyle \sum _{\text{cíclico}}S_{B}S_{C}=S^{2}\\[2pt]\displaystyle \sum _{\text{cíclico}}b^{2}c^{2}&=&S_{\omega }^{2}+S^{2}\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fcc3081f4cb0071be528558647658635fc70b95)
![{\displaystyle {\begin{aligned}K_{p}&=\sum _{\text{cíclico}}a^{2}S_{A}=2S^{2},\\[2pt]K_{q}&=\sum _{\text{cíclico}}S_{B}S_{C}=S^{2}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fdf90e1b679e0b8217e68e4dd66268877fca48c)
![{\displaystyle {\begin{aligned}D^{2}&=\sum _{\text{cíclico}}a^{2}S_{A}\left({\frac {aS_{A}}{2S^{2}}}-{\frac {S_{B}S_{C}}{aS^{2}}}\right)^{2}\\[2pt]&={\frac {1}{4S^{4}}}\sum _{\text{cíclico}}a^{4}S_{A}^{3}-{\frac {S_{A}S_{B}S_{C}}{S^{4}}}\sum _{\text{cíclico}}a^{2}S_{A}+{\frac {S_{A}S_{B}S_{C}}{S^{4}}}\sum _{\text{cíclico}}S_{B}S_{C}\\[2pt]&={\frac {1}{4S^{4}}}\sum _{\text{cíclico}}a^{2}S_{A}^{2}(S^{2}-S_{B}S_{C})-2(S_{\omega }-4R^{2})+(S_{\omega }-4R^{2})\\[2pt]&={\frac {1}{4S^{2}}}\sum _{\text{cíclico}}a^{2}S_{A}^{2}-{\frac {S_{A}S_{B}S_{C}}{S^{4}}}\sum _{\text{cíclico}}a^{2}S_{A}-(S_{\omega }-4R^{2})\\[2pt]&={\frac {1}{4S^{2}}}\sum _{\text{cíclico}}a^{2}(b^{2}c^{2}-S^{2})-{\frac {1}{2}}(S_{\omega }-4R^{2})-(S_{\omega }-4R^{2})\\[2pt]&={\frac {3a^{2}b^{2}c^{2}}{4S^{2}}}-{\frac {1}{4}}\sum _{\text{cíclico}}a^{2}-{\frac {3}{2}}(S_{\omega }-4R^{2})\\[2pt]&=3R^{2}-{\frac {1}{2}}S_{\omega }-{\frac {3}{2}}S_{\omega }+6R^{2}\\[10pt]&=9R^{2}-2S_{\omega }.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eeebdf4628f5fac61d915816d9d8918c138717eb)
