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Консультация № 192138
15.12.2017, 18:55
0.00 руб.
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Здравствуйте! Прошу помощи в следующем вопросе:
В треугольнике ABC три медианы AA1,BB1,CC1 пересекаются в точке O. Середины отрезков OA,OB,OC обозначены A2,B2,C2 соответственно. Чему равна сумма квадратов всех сторон шестиугольника A1B2C1A2B1C2, если AB=c,BC=a,AC=b?
В треугольнике ABC три медианы AA1,BB1,CC1 пересекаются в точке O. Середины отрезков OA,OB,OC обозначены A2,B2,C2 соответственно. Чему равна сумма квадратов всех сторон шестиугольника A1B2C1A2B1C2, если AB=c,BC=a,AC=b?
Обсуждение
16.12.2017, 15:25
общий
это ответ
Вектор медианы равен ma = (c - b)/2. AA2 - это треть медианы, т.е., AA2 = (c - b)/6.
Имеем AA2 + A2B1 = - b/2, AA2 + A2C1 = c/2. Отсюда получим
A2B12 = ((c - b)/6 + b/2)2 = (с/6 + b/3)2 = c2/36 + b2/9 + bc/9,
A2C12 = (c/2 - (c - b)/6)2 = (b/6 + c/3)2 = b2/36 + c2/9 + bc/9.
A2B12 + A2C12 = 5b2/36 + 5c2/36 + 2bc/9 = 5b2/36 + 5c2/36 + 2bc/9.
Возведя в квадрат b + c = a, находим 2bc = a2 - b2 - c2 и
A2B12 + A2C12 = a2/9 + b2/36 + c2/36.
Суммы квадратов двух остальных пар сторон шестиугольника получим подстановкой a -> b, b->c, c->a.
Складывая полученные выражения, найдем сумму квадратов всех сторон:
(a2/9 + b2/36 + c2/36) + (b2/9 + c2/36 + a2/36) + (c2/9 + a2/36 + b2/36) = (a2 + b2 + c2)/6.
Форма ответа
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