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Консультация № 134292
26.04.2008, 20:49
0.00 руб.
0
3
3
Здравствуйте. Доказать что у описанного четырехугольника суммы противоположных сторон равны.
Обсуждение
Неизвестный
26.04.2008, 21:34
общий
это ответ
Здравствуйте, Coolcooler1!
Пусть описанный четырехугольник ABCD, а вписанная в него окружность касается его сторон:
стороны АВ в точке N,
стороны BC в точке K,
стороны CD в точке L,
стороны AD в точке M.
Отрезки касательных, проведенных из одной точки равны, поэтому:
BK = BN
CK = CL
DL = DM
AM = AN
Тогда, BK + KC + DM + AM = BN + CL + DL + AN
BC + AD = CD + AB
что и требовалось доказать
Пусть описанный четырехугольник ABCD, а вписанная в него окружность касается его сторон:
стороны АВ в точке N,
стороны BC в точке K,
стороны CD в точке L,
стороны AD в точке M.
Отрезки касательных, проведенных из одной точки равны, поэтому:
BK = BN
CK = CL
DL = DM
AM = AN
Тогда, BK + KC + DM + AM = BN + CL + DL + AN
BC + AD = CD + AB
что и требовалось доказать
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