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МатематикаРешебник по геометрии 9 класс Мерзляк Задание 176
Задание 176
\[1)\ A_{1}A_{2}\ldots A_{n} - многоугольник;\]
\[O - центр\ окружности:\]
\[OA_{1} = OA_{2} = \ldots = OA_{n} = R,\ \ \ \]
\[A_{1}A_{2} = A_{2}A_{3} = \ldots = A_{n}A_{1};\]
\[\mathrm{\Delta}A_{1}OA_{2} = \mathrm{\Delta}A_{2}OA_{3} =\]
\[= \ldots = \mathrm{\Delta}A_{n}OA_{1};\]
\[\angle OA_{1}A_{2} = \angle OA_{2}A_{1} =\]
\[= \ldots = \angle OA_{n}A_{1};\]
\[\angle A_{1}A_{2}A_{3} = \angle A_{2}A_{3}A_{4} =\]
\[= \ldots = \angle A_{n}A_{1}A_{2}.\]
\[Ответ:\ \ да.\]
\[2)\ ABCD - прямоугольник;\ \ \ \]
\[AB \neq BC:\]
\[\angle A = \angle B = \angle C = \angle D = 90{^\circ};\]
\[\angle A + \angle C = \angle B + \angle D;\]
\[ABCD - вписанный.\]
\[Ответ:\ \ нет.\]
\[3)\ A_{1}A_{2}\ldots A_{n} - многоугольник;\]
\[O - центр\ окружности;\]
\[OH_{1}\bot A_{1}A_{2};\ \ \]
\[OH_{2}\bot A_{2}A_{3},\ \ldots;\]
\[OH_{n}\bot A_{n}A_{1}.\]
\[OH_{1} = OH_{2} = \ldots = OH_{n} = R;\]
\[A_{1}A_{2} = A_{2}A_{3} = \ldots = A_{n}A_{1};\]
\[\mathrm{\Delta}OH_{1}A_{2} = \mathrm{\Delta}OH_{2}A_{2};\ \ \ \]
\[\angle OA_{2}H_{1} = \angle OA_{2}H_{2};\]
\[\mathrm{\Delta}OA_{1}A_{2} = \mathrm{\Delta}OA_{2}A_{3};\ \ \ \]
\[OA_{1} = OA_{3};\]
\[OA_{1} = OA_{2} = \ldots = OA_{n};\]
\[\mathrm{\Delta}A_{1}OA_{2} = \mathrm{\Delta}A_{2}OA_{3} = \ldots = \mathrm{\Delta}A_{n}OA_{1};\]
\[\angle OA_{1}A_{2} = \angle OA_{2}A_{1} = \ldots = \angle OA_{n}A_{1};\]
\[\angle A_{1}A_{2}A_{3} = \angle A_{2}A_{3}A_{4} =\]
\[= \ldots = \angle A_{n}A_{1}A_{2}.\]
\[Ответ:\ \ да.\]
\[4)\ A_{1}A_{2}\ldots A_{n} - многоугольник;\]
\[O - центр\ окружности;\]
\[OH_{1}\bot A_{1}A_{2};\ \ \ \]
\[OH_{2}\bot A_{2}A_{3},\ \ldots;\ \]
\[OH_{n}\bot A_{n}A_{1}.\]
\[OH_{1} = OH_{2} = \ldots = OH_{n} = R;\]
\[\angle A_{1}A_{2}A_{3} = \angle A_{2}A_{3}A_{4} =\]
\[= \ldots = \angle A_{n}A_{1}A_{2};\]
\[\mathrm{\Delta}OH_{1}A_{2} = \mathrm{\Delta}OH_{2}A_{2};\ \ \]
\[\angle OA_{2}H_{1} = \angle OA_{2}H_{2} = \frac{1}{2}\angle A_{2};\]
\[\angle OA_{1}H_{1} = \angle OA_{2}H_{1} = \angle OA_{2}H_{2} =\]
\[= \ldots = \angle OA_{n}H_{1};\]
\[\mathrm{\Delta}A_{1}OA_{2} = \mathrm{\Delta}A_{2}OA_{3} = \ldots = \mathrm{\Delta}A_{n}OA_{1};\]
\[OA_{1} = OA_{2} = \ldots = OA_{n}.\]
\[Ответ:\ \ да.\]
