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Консультация № 201578
29.10.2021, 16:12
0.00 руб.
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Точки M и O — точка пересечения медиан и центр описанной окружности треугольника ABC соответственно, R — радиус описанной окружности, a, b, c — длины сторон треугольника. Тогда MO^2=____R^2+____*(a^2 + b^2 + c^2)
нужно вставить в пропуски ( такие вот _____ ) числа
Точки M и O — точка пересечения медиан и центр описанной окружности треугольника ABC соответственно, R — радиус описанной окружности, a, b, c — длины сторон треугольника. Тогда MO^2=____R^2+____*(a^2 + b^2 + c^2)
нужно вставить в пропуски ( такие вот _____ ) числа
Обсуждение
30.10.2021, 04:10
общий
это ответ
Согласно теореме Лейбница, для треугольника ABC, медианы которого пересекаются в точке M, и произвольной точки O имеет место равенство
Также воспользуемся тем, что сумма квадратов сторон треугольника равна утроенной сумме квадратов расстояний от точки пересечения медиан до вершин:
Тогда
и так как AO = BO = CO = R, AB = c, BC = a, CA = b, то
то есть искомые числа равны 1 и -1/9.
Также воспользуемся тем, что сумма квадратов сторон треугольника равна утроенной сумме квадратов расстояний от точки пересечения медиан до вершин:
Тогда
и так как AO = BO = CO = R, AB = c, BC = a, CA = b, то
то есть искомые числа равны 1 и -1/9.
5
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