math-public:zamechatelniye_tochki_kak_ts_mass
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math-public:zamechatelniye_tochki_kak_ts_mass [2019/05/26 23:10] – [Теорема] labreslav | math-public:zamechatelniye_tochki_kak_ts_mass [2019/05/27 12:53] – [Теорема] labreslav | ||
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Строка 136: | Строка 136: | ||
===== Теорема ===== | ===== Теорема ===== | ||
- | $$II_c=\frac{r}{\sin \frac{\alpha}{2} \cdot \sin \frac{\beta}{2}} = \sqrt{r^2+r^2_c+(p-a)^2+(p-b)^2}$$ | + | $$I_aI=\frac{r}{\sin \frac{\beta}{2} \cdot \sin \frac{\gamma}{2}} = \sqrt{r^2+r^2_a+(p-b)^2+(p-c)^2}$$ |
$$I_aI_b=r \cdot \frac{\operatorname{ctg} \frac{\alpha}{2}+\operatorname{ctg} \frac{\beta}{2}}{\sin \frac{\gamma}{2}} = \dfrac{r\cos{\frac{\gamma}{2}}}{\sin{\frac{\alpha}{2}}\sin{\frac{\beta}{2}}\sin{\frac{\gamma}{2}}}=\dfrac{r\ctg{\frac{\gamma}{2}}}{\sin{\frac{\alpha}{2}}\sin{\frac{\beta}{2}}}$$ | $$I_aI_b=r \cdot \frac{\operatorname{ctg} \frac{\alpha}{2}+\operatorname{ctg} \frac{\beta}{2}}{\sin \frac{\gamma}{2}} = \dfrac{r\cos{\frac{\gamma}{2}}}{\sin{\frac{\alpha}{2}}\sin{\frac{\beta}{2}}\sin{\frac{\gamma}{2}}}=\dfrac{r\ctg{\frac{\gamma}{2}}}{\sin{\frac{\alpha}{2}}\sin{\frac{\beta}{2}}}$$ | ||
$$\sin \frac{\alpha}{2}=\sqrt{\frac{(p-b)(p-c)}{b c}}$$ | $$\sin \frac{\alpha}{2}=\sqrt{\frac{(p-b)(p-c)}{b c}}$$ | ||
$$\operatorname{ctg} \frac{\alpha}{2}=\frac{p-a}{r}$$ | $$\operatorname{ctg} \frac{\alpha}{2}=\frac{p-a}{r}$$ | ||
- | $$II_a=a \sqrt{\frac{b c}{p(p-a)}}$$ | + | $$I_aI=a \sqrt{\frac{b c}{p(p-a)}} = \sqrt{(b-c)^2+(r+r_a)^2}$$ |
+ | $$I_{a}I_{b}=c \sqrt{\frac{a b}{(p-a)(p-b)}} = \sqrt{c^2+(r_a+r_b)^2}$$ | ||
+ | |||
+ | ===== Теорема ====== | ||
+ | |||
+ | $$O I_{a}^{2}=R^{2}+2 R r_{a}$$ | ||
+ | |||
+ | $$OI^{2}=R^{2}-2 R r$$ | ||
+ | |||
+ | ===Доказательство=== | ||
+ | |||
+ | $(a+b+c)\overrightarrow{OI} = a\overrightarrow{OA}+b\overrightarrow{OB}+c\overrightarrow{OC}$ | ||
+ | |||
+ | $4p^2\cdot OI^2 = a^2 OA^2+b^2 OB^2 + c^2 OC^2 | ||
+ | + 2ab \overrightarrow{OA}\cdot\overrightarrow{OB} | ||
+ | + 2bc \overrightarrow{OB}\cdot\overrightarrow{OC} | ||
+ | + 2ac \overrightarrow{OA}\cdot\overrightarrow{OC}$ | ||
+ | |||
+ | $4p^2\cdot OI^2 = R^2(a^2+b^2 + c^2) | ||
+ | + 2ab R^2\cos{2\gamma} | ||
+ | + 2bc R^2\cos{2\alpha} | ||
+ | + 2ac R^2\cos{2\beta}$ | ||
+ | |||
+ | $4p^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2 | ||
+ | + 2ab \cos{2\gamma} | ||
+ | + 2bc \cos{2\alpha} | ||
+ | + 2ac \cos{2\beta}\right)$ | ||
+ | |||
+ | $4p^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2 | ||
+ | + 2ab (1-2\sin^2{\gamma}) | ||
+ | + 2bc (1-2\sin^2{\alpha}) | ||
+ | + 2ac (1-2\sin^2{\beta})\right)$ | ||
+ | |||
+ | |||
+ | $4p^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2+2ab+2bc+2ac- | ||
+ | | ||
+ | + bc\sin^2{\alpha} | ||
+ | + ac\sin^2{\beta})\right)$ | ||
+ | |||
+ | |||
+ | $4p^2\cdot OI^2 = R^2\left((a+b+c)^2- | ||
+ | | ||
+ | + bc\frac{a^2}{4R^2} | ||
+ | + ac\frac{b^2}{4R^2}\right)\right)$ | ||
+ | |||
+ | $4p^2\cdot OI^2 = R^2\left(4p^2-\frac{abc}{R^2}(a+b+c)\right)$ | ||
+ | |||
+ | $4p^2\cdot OI^2 = R^2\left(4p^2-\frac{abc}{R^2}2p\right)$ | ||
+ | |||
+ | $OI^2 = R^2-\frac{abc}{2p}$ | ||
+ | |||
+ | $OI^2 = R^2-\frac{4RS}{2S/ | ||
+ | |||
+ | $OI^2 = R^2-2Rr$ | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | $(-a+b+c)\overrightarrow{OI_a} = -a\overrightarrow{OA}+b\overrightarrow{OB}+c\overrightarrow{OC}$ | ||
+ | |||
+ | $4(p-a)^2\cdot OI^2 = a^2 OA^2+b^2 OB^2 + c^2 OC^2 | ||
+ | - 2ab \overrightarrow{OA}\cdot\overrightarrow{OB} | ||
+ | + 2bc \overrightarrow{OB}\cdot\overrightarrow{OC} | ||
+ | - 2ac \overrightarrow{OA}\cdot\overrightarrow{OC}$ | ||
+ | |||
+ | $4(p-a)^2\cdot OI^2 = R^2(a^2+b^2 + c^2) | ||
+ | - 2ab R^2\cos{2\gamma} | ||
+ | + 2bc R^2\cos{2\alpha} | ||
+ | - 2ac R^2\cos{2\beta}$ | ||
+ | |||
+ | $4(p-a)^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2 | ||
+ | - 2ab \cos{2\gamma} | ||
+ | + 2bc \cos{2\alpha} | ||
+ | - 2ac \cos{2\beta}\right)$ | ||
+ | |||
+ | $4(p-a)^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2 | ||
+ | - 2ab (1-2\sin^2{\gamma}) | ||
+ | + 2bc (1-2\sin^2{\alpha}) | ||
+ | - 2ac (1-2\sin^2{\beta})\right)$ | ||
+ | |||
+ | |||
+ | $4(p-a)^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2-2ab+2bc-2ac- | ||
+ | | ||
+ | + bc\sin^2{\alpha} | ||
+ | - ac\sin^2{\beta})\right)$ | ||
+ | |||
+ | |||
+ | $4(p-a)^2\cdot OI^2 = R^2\left((-a+b+c)^2- | ||
+ | | ||
+ | + bc\frac{a^2}{4R^2} | ||
+ | - ac\frac{b^2}{4R^2}\right)\right)$ | ||
+ | |||
+ | $4(p-a)^2\cdot OI^2 = R^2\left(4(p-a)^2-\frac{abc}{R^2}(a-b-c)\right)$ | ||
+ | |||
+ | $4(p-a)^2\cdot OI^2 = R^2\left(4(p-a)^2+\frac{abc}{R^2}2(p-a)\right)$ | ||
+ | |||
+ | $OI^2 = R^2+\frac{abc}{2(p-a)}$ | ||
+ | |||
+ | $OI^2 = R^2+\frac{4RS}{2S/ | ||
+ | |||
+ | $OI^2 = R^2+2Rr_a$ | ||
+ | |||
+ | |||
math-public/zamechatelniye_tochki_kak_ts_mass.txt · Последнее изменение: 2019/05/27 15:15 — labreslav