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Консультация № 191684
07.11.2017, 23:56
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Уважаемые эксперты! Пожалуйста помогите со следующей задачей:
Если три высоты h1, h2, h3, треугольника ABC удовлетворяют условию 5(h1h2+h2h3+h3h1) = 2(h1h2h3), то чему равен радиус вписанной в треугольник ABC окружности?
Если три высоты h1, h2, h3, треугольника ABC удовлетворяют условию 5(h1h2+h2h3+h3h1) = 2(h1h2h3), то чему равен радиус вписанной в треугольник ABC окружности?
Обсуждение
08.11.2017, 06:29
общий
это ответ
Здравствуйте, Валерий!
Разделим обе части равенства, приведенного в условии задачи, на h1h2h3. Получим 1/h1 + 1/h2 + 1/h3 = 2/5.
Теперь умножим обе части на площадь треугольника S. Т.к. S = h1a/2 = h2b/2 = h3c/2,
имеем (a + b + c)/2 = 2S/5. Но S = (a + b + c)*R/2, где R - радиус вписанной окружности. Перемножая правые и левые
части этих уравнений, после сокращения найдем R = 5/2.
Разделим обе части равенства, приведенного в условии задачи, на h1h2h3. Получим 1/h1 + 1/h2 + 1/h3 = 2/5.
Теперь умножим обе части на площадь треугольника S. Т.к. S = h1a/2 = h2b/2 = h3c/2,
имеем (a + b + c)/2 = 2S/5. Но S = (a + b + c)*R/2, где R - радиус вписанной окружности. Перемножая правые и левые
части этих уравнений, после сокращения найдем R = 5/2.
5
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