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Консультация № 188934
13.03.2016, 17:51
0.00 руб.
0
3
1
Здравствуйте! Прошу помощи в следующем вопросе:
Длины сторон некоторого треугольника и диаметр вписанной в него окружности являются четырьмя натуральными числами и последовательными членами арифметической прогрессии. Максимальная длина стороны треугольника не превосходит 26. Найдите количество всех таких треугольников.
Длины сторон некоторого треугольника и диаметр вписанной в него окружности являются четырьмя натуральными числами и последовательными членами арифметической прогрессии. Максимальная длина стороны треугольника не превосходит 26. Найдите количество всех таких треугольников.
Обсуждение
13.03.2016, 20:48
общий
это ответ
Здравствуйте, Асмик Гаряка!
Начнём с того, что диаметр вписанной окружности заведомо меньше любой из сторон.
Поэтому (обозначая стороны в порядки возрастания) арифметическая прогрессия будет иметь вид d, a, b, c, где
a=d+x
b=d+2x
c=d+3x
(все числа натуральные)
воспользуемся формулой
d2=(-a+b+c)(a-b+c)(a+b-c)/(a+b+c)
d2=(d+4x)(d+2x)d/(3d+6x)
d=(d+4x)/3
3d=d+4x
2d=4x
d=2x
то есть решения имеют вид
d=2x
a=3x
b=4x
c=5x
Нетрудно заметить, что треугольник прямоугольный.
При этом, наибольшая сторона c=5x[$8804$]26
x[$8804$]5 - и мы имеем 5 натуральных значений x[$8712$]{1, 2, 3, 4, 5} и 5 наборов решений, удовлетворяющих этому условию
Начнём с того, что диаметр вписанной окружности заведомо меньше любой из сторон.
Поэтому (обозначая стороны в порядки возрастания) арифметическая прогрессия будет иметь вид d, a, b, c, где
a=d+x
b=d+2x
c=d+3x
(все числа натуральные)
воспользуемся формулой
d2=(-a+b+c)(a-b+c)(a+b-c)/(a+b+c)
d2=(d+4x)(d+2x)d/(3d+6x)
d=(d+4x)/3
3d=d+4x
2d=4x
d=2x
то есть решения имеют вид
d=2x
a=3x
b=4x
c=5x
Нетрудно заметить, что треугольник прямоугольный.
При этом, наибольшая сторона c=5x[$8804$]26
x[$8804$]5 - и мы имеем 5 натуральных значений x[$8712$]{1, 2, 3, 4, 5} и 5 наборов решений, удовлетворяющих этому условию
5
13.03.2016, 21:49
общий
Адресаты:
Я уже собрался было писать программку для перебора 
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
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