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Консультация № 187233
29.03.2013, 19:56
79.10 руб.
29.03.2013, 20:00
0
5
2
Здравствуйте! Прошу помощи в следующем вопросе: Точки А1 и В1 лежат соответственно на сторонах ВС и АС треугольника АВС и делят их в отношениях 3:1 и 1:2. Найдите отношения, в котором делятся отрезки АА1 и ВВ1 точкой их пересечения. Заранее благодарен.
Обсуждение
29.03.2013, 23:12
общий
это ответ
Здравствуйте, Алексей Валентинович!
Предложенная Вами задача решается с помощью теоремы Менелая.
Пусть |BA1|/|A1C| = 3/1, |AB1|/|B1C| = 1/2, Отрезки [AA1] и [BB1] пересекаются в точке P. Тогда по указанной теореме получаем
С уважением.
Предложенная Вами задача решается с помощью теоремы Менелая.
Пусть |BA1|/|A1C| = 3/1, |AB1|/|B1C| = 1/2, Отрезки [AA1] и [BB1] пересекаются в точке P. Тогда по указанной теореме получаем
|CA1|/|A1B| * |BP|/|PB1| *| B1A|/|AC| = 1, 1/3 * |BP|/|PB1| * 1/3=1, |BP|/|PB[sub]1[/sub]| = 9/1;
|CB1|/|B1A| * |AP|/|PA1| * |A1B|/|BC| = 1, 2/1 * |AP|/|PA1| * 3/4 = 1, |AP|/|PA[sub]1[/sub]| = 2/3.
С уважением.
5
Об авторе:
Facta loquuntur.
Facta loquuntur.
30.03.2013, 00:48
общий
Адресаты:
Интересная задачка и изящное решение 
Об авторе:
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
30.03.2013, 14:10
общий
это ответ
Здравствуйте, Тимофеев Алексей Валентинович!
Изящное решение получается как раз без использования теоремы Менелая. Проведем отрезок A1B2, параллельный ВВ1 (B2 лежит на AC). Тогда из подобия треугольников BB1C и B2A1C получаем, что В1В2:B2C=3:1, поэтому В1В2=(3/4)В1С. А так как AВ1=(1/2)В1C, то AВ1=(1/2)(4/3)BВ2=(2/3)В1В2.
Если P - точка пересечения прямых BВ1 и AA1, то из подобия треугольников AA1В2 и APВ1 следует, что AP:PA1=AВ1:В1В2=2:3.
Аналогично находится и второе отношение.
Изящное решение получается как раз без использования теоремы Менелая. Проведем отрезок A1B2, параллельный ВВ1 (B2 лежит на AC). Тогда из подобия треугольников BB1C и B2A1C получаем, что В1В2:B2C=3:1, поэтому В1В2=(3/4)В1С. А так как AВ1=(1/2)В1C, то AВ1=(1/2)(4/3)BВ2=(2/3)В1В2.
Если P - точка пересечения прямых BВ1 и AA1, то из подобия треугольников AA1В2 и APВ1 следует, что AP:PA1=AВ1:В1В2=2:3.
Аналогично находится и второе отношение.
5
01.04.2013, 00:05
общий
Адресаты:
Здравствуйте, Дмитрий Германович!
Ваше решение задачи мне нравится. Правда, оно требует дополнительных построений и выкладок и поэтому вряд ли может быть более "изящным", чем решение с использованием теоремы Менелая. Другое дело, что теорема Менелая не рассматривается в школьном курсе геометрии и поэтому Ваше решение предпочтительнее, как я думаю. 
В любом случае мне понравился Ваш подход. Я его как-то упустил из виду.
С уважением.
Ваше решение задачи мне нравится.
Правда, оно требует дополнительных построений и выкладок и поэтому вряд ли может быть более "изящным", чем решение с использованием теоремы Менелая. Другое дело, что теорема Менелая не рассматривается в школьном курсе геометрии и поэтому Ваше решение предпочтительнее, как я думаю. В любом случае мне понравился Ваш подход. Я его как-то упустил из виду.
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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