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math-public:rasstojanija_oi_oia

Теорема

$$OI^{2}=R^{2}-2 R r$$

Доказательство

$(a+b+c)\overrightarrow{OI} = a\overrightarrow{OA}+b\overrightarrow{OB}+c\overrightarrow{OC}$

$4p^2\cdot OI^2 = a^2 OA^2+b^2 OB^2 + c^2 OC^2 + 2ab \overrightarrow{OA}\cdot\overrightarrow{OB} + 2bc \overrightarrow{OB}\cdot\overrightarrow{OC} + 2ac \overrightarrow{OA}\cdot\overrightarrow{OC}$

$4p^2\cdot OI^2 = R^2(a^2+b^2 + c^2) + 2ab R^2\cos{2\gamma} + 2bc R^2\cos{2\alpha} + 2ac R^2\cos{2\beta}$

$4p^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2 + 2ab \cos{2\gamma} + 2bc \cos{2\alpha} + 2ac \cos{2\beta}\right)$

$4p^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2 + 2ab (1-2\sin^2{\gamma}) + 2bc (1-2\sin^2{\alpha}) + 2ac (1-2\sin^2{\beta})\right)$

$4p^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2+2ab+2bc+2ac- 4(ab\sin^2{\gamma} + bc\sin^2{\alpha} + ac\sin^2{\beta})\right)$

$4p^2\cdot OI^2 = R^2\left((a+b+c)^2- 4\left(ab\frac{c^2}{4R^2} + bc\frac{a^2}{4R^2} + ac\frac{b^2}{4R^2}\right)\right)$

$4p^2\cdot OI^2 = R^2\left(4p^2-\frac{abc}{R^2}(a+b+c)\right)$

$4p^2\cdot OI^2 = R^2\left(4p^2-\frac{abc}{R^2}2p\right)$

$OI^2 = R^2-\frac{abc}{2p}$

$OI^2 = R^2-\frac{4RS}{2S/r}$

$OI^2 = R^2-2Rr$

Теорема

$$O I_{a}^{2}=R^{2}+2 R r_{a}$$

Доказательство

$(-a+b+c)\overrightarrow{OI_a} = -a\overrightarrow{OA}+b\overrightarrow{OB}+c\overrightarrow{OC}$

$4(p-a)^2\cdot OI^2 = a^2 OA^2+b^2 OB^2 + c^2 OC^2 - 2ab \overrightarrow{OA}\cdot\overrightarrow{OB} + 2bc \overrightarrow{OB}\cdot\overrightarrow{OC} - 2ac \overrightarrow{OA}\cdot\overrightarrow{OC}$

$4(p-a)^2\cdot OI^2 = R^2(a^2+b^2 + c^2) - 2ab R^2\cos{2\gamma} + 2bc R^2\cos{2\alpha} - 2ac R^2\cos{2\beta}$

$4(p-a)^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2 - 2ab \cos{2\gamma} + 2bc \cos{2\alpha} - 2ac \cos{2\beta}\right)$

$4(p-a)^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2 - 2ab (1-2\sin^2{\gamma}) + 2bc (1-2\sin^2{\alpha}) - 2ac (1-2\sin^2{\beta})\right)$

$4(p-a)^2\cdot OI^2 = R^2\left(a^2+b^2 + c^2-2ab+2bc-2ac- 4(-ab\sin^2{\gamma} + bc\sin^2{\alpha} - ac\sin^2{\beta})\right)$

$4(p-a)^2\cdot OI^2 = R^2\left((-a+b+c)^2- 4\left(-ab\frac{c^2}{4R^2} + bc\frac{a^2}{4R^2} - ac\frac{b^2}{4R^2}\right)\right)$

$4(p-a)^2\cdot OI^2 = R^2\left(4(p-a)^2-\frac{abc}{R^2}(a-b-c)\right)$

$4(p-a)^2\cdot OI^2 = R^2\left(4(p-a)^2+\frac{abc}{R^2}2(p-a)\right)$

$OI^2 = R^2+\frac{abc}{2(p-a)}$

$OI^2 = R^2+\frac{4RS}{2S/r_a}$

$OI^2 = R^2+2Rr_a$

math-public/rasstojanija_oi_oia.txt · Последнее изменение: 2019/05/27 15:14 — labreslav

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