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Консультация № 189680
11.08.2016, 12:33
0.00 руб.
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Здравствуйте! Прошу помощи в следующем вопросе ( №10 - а,б ):
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Обсуждение
11.08.2016, 15:46
общий
это ответ
Здравствуйте, anton74551!
Представлю даже 2 варианта доказательства
Первый вариант - по подсказке в условии:
Строим квадрат CKMN со стороной CK=AC+BC, помещаем [$8895$]ACB в один из углов и заполняем остальные такими же треугольниками.
Убеждаемся (например из симметрии при повороте относительно центра квадрата на 90[$176$]), что гипотенузы образуют квадрат ADFB, а его центр O совпадает с центром квадрата CKMN.
Следовательно, CO лежит на диагонали CM квадрата CKMN и является биссектрисой угла [$8736$]C
Второй вариант - с помощью описанной окружности:
Строим описанную вокруг [$8895$]ABC окружность (гипотенуза AC является диаметром) и убеждаемся (например, из того, что [$8895$]AOB - прямоугольный равнобедренный), что точка точка O лежит на этой окружности и разделяет дуги [$8745$]AO=[$8745$]OB=90[$176$]
Как опирающиеся на эти дуги вписанные углы, [$8736$]ACO=[$8736$]BCO=45[$176$], что и требовалось доказать
Представлю даже 2 варианта доказательства
Первый вариант - по подсказке в условии:
Строим квадрат CKMN со стороной CK=AC+BC, помещаем [$8895$]ACB в один из углов и заполняем остальные такими же треугольниками.
Убеждаемся (например из симметрии при повороте относительно центра квадрата на 90[$176$]), что гипотенузы образуют квадрат ADFB, а его центр O совпадает с центром квадрата CKMN.
Следовательно, CO лежит на диагонали CM квадрата CKMN и является биссектрисой угла [$8736$]C
Второй вариант - с помощью описанной окружности:
Строим описанную вокруг [$8895$]ABC окружность (гипотенуза AC является диаметром) и убеждаемся (например, из того, что [$8895$]AOB - прямоугольный равнобедренный), что точка точка O лежит на этой окружности и разделяет дуги [$8745$]AO=[$8745$]OB=90[$176$]
Как опирающиеся на эти дуги вписанные углы, [$8736$]ACO=[$8736$]BCO=45[$176$], что и требовалось доказать
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