Решебник по геометрии 9 класс Мерзляк Задание 169

\[Схематический\ рисунок.\]

\[Дано:\]

\[O - центр\ впис.\ окружности;\]

\[AA_{1},\ BB_{1},\ CC_{1} - высоты\ \mathrm{\Delta}ABC;\]

\[AA_{1} = h_{1};\]

\[BB_{1} = h_{2};\]

\[CC_{1} = h_{3}.\]

\[Доказать:\]

\[\frac{1}{h_{1}} + \frac{1}{h_{2}} + \frac{1}{h_{3}} = \frac{1}{r}.\]

\[Доказательство.\]

\[1)\ Пусть:\]

\[BC = a,\ \ \ \]

\[AC = b,\ \ \ \]

\[AB = c.\]

\[2)\ В\ \mathrm{\Delta}ABC:\]

\[S = \frac{1}{2}ah_{1} = \frac{1}{2}bh_{2} = \frac{1}{2}ch_{3};\]

\[S = pr = \frac{1}{2}(a + b + c)\text{r.}\]

\[ah_{1} = bh_{2} = ch_{3} = (a + b + c)r;\]

\[\frac{a}{a + b + c} = \frac{r}{h_{1}};\]

\[\frac{b}{a + b + c} = \frac{r}{h_{2}};\]

\[\frac{c}{a + b + c} = \frac{r}{h_{3}};\]

\[\frac{r}{h_{1}} + \frac{r}{h_{2}} + \frac{r}{h_{3}} = \frac{a + b + c}{a + b + c} = 1\]

\[\frac{1}{h_{1}} + \frac{1}{h_{2}} + \frac{1}{h_{3}} = \frac{1}{r}.\]

\[Что\ и\ требовалось\ доказать.\]