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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Какая наименьшая площадь может быть у треуг. AOC, вершина O которого лежит на катете BC прямоугольного треуг. ABC и является центром окружности радиуса R, касающейся гипотенузы AC и проходящей через точку B?
Так как окружность касается сторон угла ВСА, то СО – биссектриса BCA .Пусть уг BCA= 2а . Тогда BC=Rctgа и AB=BCtg2а Дальше вычислить площадь и найти наименьшее значение через производную
Выполните рисунок и Вы увидите, что в соответствии с условием задачи окружность с центром O на катете BC касается не сторон угла BCA, а сторон угла BAC. Не CO -- биссектриса угла BCA, а AO -- биссектриса угла BAC...
Можно и подробнее. Можно даже получить готовое решение. Но мне не хочется лишать Вас возможности решить задачу самостоятельно. Иначе зачем Вам задания повышенного уровня сложности от МФТИ?
Я продлю срок действия консультации на пять суток, чтобы у Вас было время поразмышлять и обсудить свои мысли.
Вы видите на рисунке указанные мной пары треугольников? Давайте обозначим точку, в которой окружность касается стороны AC, буквой D...
Обозначим точку пересечения отрезка и окружности буквой Учитывая равенство треугольников и а также подобие треугольников и мы получим выражение для площади треугольника
Используя методы дифференциального исчисления, мы можем вычислить наименьшее значение этой площади:
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lk 
и окружности буквой 
Учитывая равенство треугольников 
и 
а также подобие треугольников 
и 
мы получим выражение для площади 
треугольника 
и приравняйте её к нулю.
