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Консультация № 196044
03.08.2019, 22:16
0.00 руб.
0
4
2
Здравствуйте! Прошу помощи в следующем вопросе:
Пусть BB1 высота треугольника ABC; O - центр его опи-
санной окружности; B2 точка пересечения прямой BB1 с окружностью,
отличная от B; H-точка пересечения высот треугольника. Докажите:
а) HB1= B1B2 (т. е. точка, симметричная ортоцентру треугольника
относительно его стороны, лежит на окружности, описанной около тре-
угольника);
б) расстояние от O до AC вдвое меньше, чем BH;
в) Угол ABH= углу OBC. Пункт а) и б) доказала, помогите с в).
Пусть BB1 высота треугольника ABC; O - центр его опи-
санной окружности; B2 точка пересечения прямой BB1 с окружностью,
отличная от B; H-точка пересечения высот треугольника. Докажите:
а) HB1= B1B2 (т. е. точка, симметричная ортоцентру треугольника
относительно его стороны, лежит на окружности, описанной около тре-
угольника);
б) расстояние от O до AC вдвое меньше, чем BH;
в) Угол ABH= углу OBC. Пункт а) и б) доказала, помогите с в).
Обсуждение
05.08.2019, 06:06
общий
это ответ
Здравствуйте, Mari!
.
Из центра описанной окружности точки О опустим на ВС перпендикуляр.
.
Треугольник ВОС равнобедренный, так как ВО и СО - радиусы описанной окружности. Поэтому ОМ - высота, медиана, биссектриса в этом треугольнике.
Угол ВОС - центральной - равен дуге, на которую опирается, а угол ВОМ равен половине центрального угла, значит равен половине дуги ВС
Значит угол ВОМ равен углу ВАС. Значит треугольники АВН и ВОМ подобны. Значит угол АВН = углу СВО
.
Из центра описанной окружности точки О опустим на ВС перпендикуляр.
.Треугольник ВОС равнобедренный, так как ВО и СО - радиусы описанной окружности. Поэтому ОМ - высота, медиана, биссектриса в этом треугольнике.
Угол ВОС - центральной - равен дуге, на которую опирается, а угол ВОМ равен половине центрального угла, значит равен половине дуги ВС
Значит угол ВОМ равен углу ВАС. Значит треугольники АВН и ВОМ подобны. Значит угол АВН = углу СВО
5
05.08.2019, 08:37
общий
это ответ
Здравствуйте, Mari!
Еще одно решение.
[$916$]BB'C прямоугольный, [$8736$]BAC =[$8736$] BB'C, т.к. опираются на одну дугу BC.
Следовательно, [$916$]ABB1 и [$916$]BCB' подобны и [$8736$]ABH = [$8736$]OBC.
Еще одно решение.
[$916$]BB'C прямоугольный, [$8736$]BAC =[$8736$] BB'C, т.к. опираются на одну дугу BC.
Следовательно, [$916$]ABB1 и [$916$]BCB' подобны и [$8736$]ABH = [$8736$]OBC.
5
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