Задание 5, стр 60

Question

Через стороны АС треугольника ABC проведена плоскость $\alpha$, а из вер
шины B к этой плоскости проведен перпендикуляр BB1. Если двугранный
угол между плоскостью треугольника и плоскостью $\alpha$ равен $\phi$, покажите, что $S_{AB_1C}=S_{ABC}\cdot \cos \phi.$
ewline
	extbf{Указание.} Используйте отношения $S_{ABC}=\frac{1}{2}AC\cdot BD; \ B_1D = BD\cdot \cos \phi$.

Gimli · Accepted Answer

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Из формул в указании, можем заключить, что
$$S_{ABC}=\frac{1}{2}AC\cdot BD; \ BD=\frac{B_1D}{\cos \phi}\implies S_{ABC}=\frac{1}{2}AC\cdot \frac{B_1D}{\cos \phi}$$
Также, имеем
$$S_{ABC}\cdot \cos \phi = \frac{1}{2}AC\cdot \frac{B_1D}{\cos \phi}\cdot \cos \phi = \frac{1}{2}DB_1\cdot AC$$
$$\implies S_{ABC}\cdot \cos \phi=S_{AB_1C},$$
что и требовалось доказать. $\Box$

Задание 5, стр 60

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Задание 5, стр 60

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