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--- math-public:zamechatelniye_tochki_kak_ts_mass [2019/05/26 22:25] – [Теорема] labreslav
+++ math-public:zamechatelniye_tochki_kak_ts_mass [2019/05/27 15:15] (текущий) – labreslav
@@ Строка 132: / Строка 132: @@
 $I_aZ^2 = \sqrt{r_a^2-\dfrac13 (p-a)^2+\dfrac29 (a^2+b^2+c^2)}$
-$IZ^2 = \dfrac19\sqrt{9r_a^2-3(p-a)^2+2 (a^2+b^2+c^2)}$
+$I_aZ^2 = \dfrac19\sqrt{9r_a^2-3(p-a)^2+2 (a^2+b^2+c^2)}$
 ===== Теорема =====
-$$II_c=\frac{r}{\sin \frac{\alpha}{2} \cdot \sin \frac{\beta}{2}} = \sqrt{r^2+r^2_c+(p-a)^2+(p-b)^2}$$
+$$I_aI=\frac{r}{\sin \frac{\beta}{2} \cdot \sin \frac{\gamma}{2}} = \sqrt{r^2+r^2_a+(p-b)^2+(p-c)^2}$$
 $$I_aI_b=r \cdot \frac{\operatorname{ctg} \frac{\alpha}{2}+\operatorname{ctg} \frac{\beta}{2}}{\sin \frac{\gamma}{2}} = \dfrac{r\cos{\frac{\gamma}{2}}}{\sin{\frac{\alpha}{2}}\sin{\frac{\beta}{2}}\sin{\frac{\gamma}{2}}}=\dfrac{r\ctg{\frac{\gamma}{2}}}{\sin{\frac{\alpha}{2}}\sin{\frac{\beta}{2}}}$$
+$$\sin \frac{\alpha}{2}=\sqrt{\frac{(p-b)(p-c)}{b c}}$$
+$$\operatorname{ctg} \frac{\alpha}{2}=\frac{p-a}{r}$$
+$$I_aI=a \sqrt{\frac{b c}{p(p-a)}} = \sqrt{(b-c)^2+(r+r_a)^2}$$
+$$I_{a}I_{b}=c \sqrt{\frac{a b}{(p-a)(p-b)}} = \sqrt{c^2+(r_a+r_b)^2}$$
+===== Теорема ======
+$$O I_{a}^{2}=R^{2}+2 R r_{a}$$
+$$OI^{2}=R^{2}-2 R r$$
+===Доказательство===
+$(a+b+c)\overrightarrow{OI} = a\overrightarrow{OA}+b\overrightarrow{OB}+c\overrightarrow{OC}$
+$4p^2\cdot OI^2 = a^2 OA^2+b^2 OB^2 + c^2 OC^2
+ + 2ab \overrightarrow{OA}\cdot\overrightarrow{OB}
+ + 2bc \overrightarrow{OB}\cdot\overrightarrow{OC}
+ + 2ac \overrightarrow{OA}\cdot\overrightarrow{OC}$
+$4p^2\cdot OI^2 = R^2(a^2+b^2 + c^2)
+ + 2ab R^2\cos{2\gamma}
+ + 2bc R^2\cos{2\alpha}
+ + 2ac R^2\cos{2\beta}$
+$4p^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2
+ + 2ab \cos{2\gamma}
+ + 2bc \cos{2\alpha}
+ + 2ac \cos{2\beta}\right)$
+$4p^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2
+ + 2ab (1-2\sin^2{\gamma})
+ + 2bc (1-2\sin^2{\alpha})
+ + 2ac (1-2\sin^2{\beta})\right)$
+$4p^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2+2ab+2bc+2ac-
+(ab\sin^2{\gamma}
+ + bc\sin^2{\alpha}
+ + ac\sin^2{\beta})\right)$
+$4p^2\cdot OI^2 = R^2\left((a+b+c)^2-
+\left(ab\frac{c^2}{4R^2}
+ + bc\frac{a^2}{4R^2}
+ + ac\frac{b^2}{4R^2}\right)\right)$
+$4p^2\cdot OI^2 = R^2\left(4p^2-\frac{abc}{R^2}(a+b+c)\right)$
+$4p^2\cdot OI^2 = R^2\left(4p^2-\frac{abc}{R^2}2p\right)$
+$OI^2 = R^2-\frac{abc}{2p}$
+$OI^2 = R^2-\frac{4RS}{2S/r}$
+$OI^2 = R^2-2Rr$
+$(-a+b+c)\overrightarrow{OI_a} = -a\overrightarrow{OA}+b\overrightarrow{OB}+c\overrightarrow{OC}$
+$4(p-a)^2\cdot OI^2 = a^2 OA^2+b^2 OB^2 + c^2 OC^2
+ - 2ab \overrightarrow{OA}\cdot\overrightarrow{OB}
+ + 2bc \overrightarrow{OB}\cdot\overrightarrow{OC}
+ - 2ac \overrightarrow{OA}\cdot\overrightarrow{OC}$
+$4(p-a)^2\cdot OI^2 = R^2(a^2+b^2 + c^2)
+ - 2ab R^2\cos{2\gamma}
+ + 2bc R^2\cos{2\alpha}
+ - 2ac R^2\cos{2\beta}$
+$4(p-a)^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2
+ - 2ab \cos{2\gamma}
+ + 2bc \cos{2\alpha}
+ - 2ac \cos{2\beta}\right)$
+$4(p-a)^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2
+ - 2ab (1-2\sin^2{\gamma})
+ + 2bc (1-2\sin^2{\alpha})
+ - 2ac (1-2\sin^2{\beta})\right)$
+$4(p-a)^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2-2ab+2bc-2ac-
+(-ab\sin^2{\gamma}
+ + bc\sin^2{\alpha}
+ - ac\sin^2{\beta})\right)$
+$4(p-a)^2\cdot OI^2 = R^2\left((-a+b+c)^2-
+\left(-ab\frac{c^2}{4R^2}
+ + bc\frac{a^2}{4R^2}
+ - ac\frac{b^2}{4R^2}\right)\right)$
+$4(p-a)^2\cdot OI^2 = R^2\left(4(p-a)^2-\frac{abc}{R^2}(a-b-c)\right)$
+$4(p-a)^2\cdot OI^2 = R^2\left(4(p-a)^2+\frac{abc}{R^2}2(p-a)\right)$
+$OI^2 = R^2+\frac{abc}{2(p-a)}$
+$OI^2 = R^2+\frac{4RS}{2S/r_a}$
+$OI^2 = R^2+2Rr_a$