Решебник по геометрии 11 класс. Атанасян ФГОС 793

\[\boxed{\mathbf{793.}ОК\ ГДЗ\ –\ домашка\ на\ 5}\]

\[Дано:\ \]

\[ABCD - тетраэдр;\ \ \]

\[AD = BD = CD = a;\ \ \]

\[DH - прямая;\]

\[DH \cap ABC = H;\ \ \ \]

\[\angle ADH = \angle BDH = \angle CDH = \varphi.\]

\[Доказать:\ \]

\[DH\bot ABC.\]

\[Доказательство.\]

\[1)\ Обозначим\ ребра\ тетраэдра,\ \]

\[как\ векторы:\]

\[\overrightarrow{\text{DA}} \bullet \overrightarrow{\text{DH}} = \overrightarrow{\text{DB}} \bullet \overrightarrow{\text{DH}} = \overrightarrow{\text{DC}} \bullet \overrightarrow{\text{DH}} =\]

\[= a \bullet DH \bullet \cos\varphi\ \]

\[\left( \overrightarrow{\text{DB}} - \overrightarrow{\text{DA}} \right) \bullet \overrightarrow{\text{DH}} =\]

\[= \left( \overrightarrow{\text{DC}} - \overrightarrow{\text{DA}} \right) \bullet \overrightarrow{\text{DH}} = 0\]

\[\overrightarrow{\text{AB}} \bullet \overrightarrow{\text{DH}} = \overrightarrow{\text{AC}} \bullet \overrightarrow{\text{DH}} = 0.\ \]

\[2)\ Таким\ образом:\ \]

\[\overrightarrow{\text{AB}} \bullet \overrightarrow{\text{DH}} = 0\ \ и\ \ \overrightarrow{\text{AC}}\mathbf{\bullet}\overrightarrow{\text{DH}} = 0.\]

\[Следовательно:\]

\[DH\bot AB;\ DH\bot AC;\]

\[AH\bot ABC.\]

\[Что\ и\ требовалось\ доказать\mathbf{.}\]