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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Доказать теорему Птолемея: если четырёхугольник ABCD вписан в окружность, то имеет место равенство AB*CD +AD*BC= AC *BD . Применить эту теорему к решению задачи: квадрат ABCD вписан в окружность, на дуге ВС, не содержащей точек А и D, взята точка М. Найти отношение (AM+MD):(MC+MB)
Вы сделаете самостоятельно нужные рисунки, а я отвечу на Ваш вопрос, точнее, на два вопроса. Цитата: elenatylkevich
Прошу вас ответить на следующий вопрос: Доказать теорему Птолемея: если четырёхугольник ABCD вписан в окружность, то имеет место равенство AB*CD +AD*BC= AC *BD .
Проведём из вершины отрезок до пересечения с отрезком в точке так, что Углы и вписанные в окружность, опираются на одну и ту же дугу Поэтому Тогда треугольники и подобны по двум углам. При этом
Аналогичным образом подобны треугольники и у которых При этом
Применить эту теорему к решению задачи: квадрат ABCD вписан в окружность, на дуге ВС, не содержащей точек А и D, взята точка М. Найти отношение (AM+MD):(MC+MB)
В квадрате
В четырёхугольнике
В четырёхугольнике
Сложив равенства и и сократив на получим
Прежде чем списывать решение задачи, тщательно проверьте его во избежание ошибок.
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elenatylkevich 
отрезок до пересечения с отрезком 
в точке 
так, что 
Углы 
и 
вписанные в окружность, опираются на одну и ту же дугу 
Поэтому 
Тогда треугольники 
и 
подобны по двум углам. При этом
и 
у которых 
При этом
и 
почленно, получим
и 
и сократив на 
получим
