Различия

Показаны различия между двумя версиями страницы.

--- math-public:otrtocentr [2019/05/26 23:30] – labreslav
+++ math-public:otrtocentr [2019/05/26 23:33] – [Лемма 3] labreslav
@@ Строка 493: / Строка 493: @@
 $-2ab\overrightarrow{HA}\cdot\overrightarrow{HB}+2bc\overrightarrow{HB}\cdot\overrightarrow{HC}-2ac\overrightarrow{HA}\cdot\overrightarrow{HC} = $
-$ = +2a^2b^2\ctg^2{\alpha}\ctg^2{\beta}\cos{(180^\circ-\gamma)} + 2b^2c^2\ctg^2{\beta}\ctg^2{\gamma}\cos{(180^\circ-\alpha)} + 2a^2c^2\ctg^2{\alpha}\ctg^2{\gamma}\cos{(180^\circ-\beta)} = $
+$ = -2a^2b^2\ctg^2{\alpha}\ctg^2{\beta}\cos{(180^\circ-\gamma)} + 2b^2c^2\ctg^2{\beta}\ctg^2{\gamma}\cos{(180^\circ-\alpha)} - 2a^2c^2\ctg^2{\alpha}\ctg^2{\gamma}\cos{(180^\circ-\beta)} = $
-$ = -2a^2b^2\ctg^2{\alpha}\ctg^2{\beta}\cos{\gamma} - 2b^2c^2\ctg^2{\beta}\ctg^2{\gamma}\cos{\alpha} - 2a^2c^2\ctg^2{\alpha}\ctg^2{\gamma}\cos{\beta} = $
+$ = 2a^2b^2\ctg^2{\alpha}\ctg^2{\beta}\cos{\gamma} - 2b^2c^2\ctg^2{\beta}\ctg^2{\gamma}\cos{\alpha} 2a^2c^2\ctg^2{\alpha}\ctg^2{\gamma}\cos{\beta} = $
-$ = -2\cos{\alpha}\cos{\beta}\cos{\gamma}\left(\dfrac{a^2b^2}{\sin{\alpha}\sin{\beta}}+\dfrac{b^2c^2}{\sin{\beta}\sin{\gamma}}+\dfrac{a^2c^2}{\sin{\alpha}\sin{\gamma}}\right) = $
+$ = -2\cos{\alpha}\cos{\beta}\cos{\gamma}\left(-\dfrac{a^2b^2}{\sin{\alpha}\sin{\beta}}+\dfrac{b^2c^2}{\sin{\beta}\sin{\gamma}}-\dfrac{a^2c^2}{\sin{\alpha}\sin{\gamma}}\right) = $
-$ = -2\cos{\alpha}\cos{\beta}\cos{\gamma}\left(4R^2ab+4R^2bc+4R^2ac\right) = $
+$ = -2\cos{\alpha}\cos{\beta}\cos{\gamma}\left(-4R^2ab+4R^2bc-4R^2ac\right) = $
-$ = -2\left(\dfrac{a^2+b^2+c^2}{8R^2}-1\right)4R^2\left(ab+bc+ac\right) = $
+$ = -2\left(\dfrac{a^2+b^2+c^2}{8R^2}-1\right)4R^2\left(-ab+bc-ac\right) = $
-$ = 4R^2\left(2ab+2bc+2ac\right)-(a^2+b^2+c^2)(ab+bc+ac)$
+$ = 4R^2\left(-2ab+2bc-2ac\right)-(a^2+b^2+c^2)(-ab+bc-ac)$