Решебник по геометрии 8 класс Мерзляк Задание 499

\[\mathrm{\Delta}A_{1}B_{1}C_{1}\sim\mathrm{\Delta}A_{2}B_{2}C_{2}\sim\mathrm{\Delta}A_{3}B_{3}C_{3};\ \ \ \]

\[A_{1}B_{1} > B_{1}C_{1} > A_{1}C_{1}.\]

\[1)\ \frac{A_{1}B_{1}}{A_{2}B_{2}} = \frac{A_{1}C_{1}}{A_{2}C_{2}},\ \ \ \frac{A_{1}B_{1}}{A_{1}C_{1}} = \frac{A_{2}B_{2}}{A_{2}C_{2}};\]

\[\frac{A_{2}B_{2}}{A_{3}B_{3}} = \frac{A_{2}C_{2}}{A_{3}C_{3}},\ \ \ \frac{A_{2}B_{2}}{A_{2}C_{2}} = \frac{A_{3}B_{3}}{A_{3}C_{3}};\]

\[A_{1}B_{1}\ :A_{1}C_{1} = A_{2}B_{2}\ :A_{2}C_{2} = A_{3}B_{3}\ :A_{3}C_{3}.\]

\[2)\ Треугольники\ MNK\ и\ DEF:\]

\[MN = A_{1}B_{1};\ \ \ \]

\[NK = A_{2}B_{2};\ \ \ \]

\[MK = A_{3}B_{3};\]

\[DE = A_{1}C_{1};\]

\[EF = A_{2}C_{2};\]

\[DF = A_{3}C_{3}.\]

\[3)\ \mathrm{\Delta}MNK\sim\mathrm{\Delta}DEF - третий\ \]

\[признак:\]

\[\frac{\text{MN}}{\text{DE}} = \frac{A_{1}B_{1}}{A_{1}C_{1}};\]

\[\frac{\text{NK}}{\text{EF}} = \frac{A_{2}B_{2}}{A_{2}C_{2}};\]

\[\frac{\text{MK}}{\text{DF}} = \frac{A_{3}B_{3}}{A_{3}C_{3}};\]

\[\frac{\text{MN}}{\text{DE}} = \frac{\text{NK}}{\text{EF}} = \frac{\text{MK}}{\text{DF}}.\]

\[Ответ:\ \ да.\]