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Консультация № 194334
03.01.2019, 12:01
0.00 руб.
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2
1
Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
В треугольник АВС вписан треугольник PQR так, что Р лежит на АВ, Q – на ВС, R – на АС. Докажите, что площадь хотя бы одного из треугольников APR, BPQ и CRQ не превосходит ¼ площади всего треугольника АВС.
В треугольник АВС вписан треугольник PQR так, что Р лежит на АВ, Q – на ВС, R – на АС. Докажите, что площадь хотя бы одного из треугольников APR, BPQ и CRQ не превосходит ¼ площади всего треугольника АВС.
Обсуждение
03.01.2019, 12:14
общий
Обращаю внимание экспертов, что эта задача входит в программу заочного тура олимпиады по математике здесь. Этот тур продлён до 5 января 2019 года. Поэтому прошу желающих решить задачу не показывать её решения до 6 января.
Об авторе:
Facta loquuntur.
Facta loquuntur.
06.01.2019, 15:13
общий
это ответ
Здравствуйте, alyona74@mail.ru!
Пусть
Тогда 
При этом 
По теореме об отношении площадей треугольников, имеющих по равному углу при вершине,
Если все эти отношения больше
то совместной является система неравенств
Тогда должно быть истинным неравенство
Но последнее неравенство является ложным, потому что
Значит, хотя бы одно из отношений
должно быть не больше 
Пусть
Тогда При этом По теореме об отношении площадей треугольников, имеющих по равному углу при вершине,
Если все эти отношения больше
то совместной является система неравенствТогда должно быть истинным неравенство
Но последнее неравенство является ложным, потому что
Значит, хотя бы одно из отношений
должно быть не больше
Об авторе:
Facta loquuntur.
Facta loquuntur.
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