?9?33(3cos??sin?)9?33sin?9??

4cos?4cos?4333?3sin?33xM??，其中0????120?或180????240? 34cos?4?yM??9?33sin?933?3sin?33???33?,??3,0? ?O(0,0)，M??，N?4cos?44cos?4(3cos??sin?)?????3?2xM33?3sin?33222|OM|?xM?yM?xM??xM?????2yM

32cos?23??|ON|?33?3?xN?xN

3cos??sin?222?9?33sin?333??33?3sin?33?|MN|??????????4?3cos??sin?4??4cos??4cos?2?xM?xN?222?yM

?(9?33sin?)(3cos??sin?)?33?4cos?3??33?3sin?33?????????

44cos?44cos?(3cos??sin?)????由于△OMN为等腰三角形

①当|OM|?|ON|时?2yM?xN?xN?2 yM33?33cos??sin???2?33?3sin?33?4cos?43?3cos??sin?3cos??sin??2?3?3cos??sin?4cos?3?13cos??sin??2 3?sin?3?4cos?44cos?cos?1?2?? ?22sin??603cos??sin?????sin???60???|cos?|?|3cos??sin?|2?(2|cos?|)2 ?3cos2??sin2??23sin?cos??4cos2? ?3sin2??cos2??sin2??cos2?

?tan2??13（0????120?或180????240?或300????360?） ?33?2??30?或2??180??30??210?

???15?或??105?

?92?36?3???15???3333?2?3??3?经时|ON|?XN? ??2sin???60?3cos??sin??92?36?3??105?????2（2）当|OM|?|MN|时?2yM??3yM?xN?22?yM?xN?23yM?xN?23 yM?4cos2??9cos2??3sin2??63sin?cos?

?4?1?cos2?1?cos2?1?cos2??9??3??33sin2? 222?cos2??33sin2???4（0????120?或180????240?或300????360?）

，或??330?或??289.1066054????（舍） ???109.1066954????或??150?（舍）

???109.1066054?L或??330?

当??109.1066054?时，

?4cos??23?2cos??3cos??3sin?

3cos??sin?33339?3??3??3?0，舍去

2cos?2cos?3cos??tan?3此时|ON|?xN?也可用2cos??0但3cos??3sin??0来得出矛值！ ∴??330? 此时|ON|?xN?3333?3??3?3?33 2sin???60??3cosx?sin?③当|ON|?|MN|时，xN??3yM?xN?22?yM?xN2? yM3?4cos?2??23cos??3cos??sin?

3cos??sin?3?12cos2??3cos2??sin2??23sin?cos?

?12?1?cos2?1?cos2?1?cos2??3???3sin2? 222

?3sin2??5cos2??4（0????120?或180????240?或300????360?）

． ???60?或??100.8933946????或??240?（舍）或??280.8933946????（舍）当??100.8933946????时，23cos??0，3cos??sin??0舍去．

???60?

此时|ON|?x3N?33cos??sin??3?332sin???60???3?6

综上所述，当△OMN为等腰三角形时， 线段ON的长为

92?362?3???15??或6???60??3?33???330??．

或92?362?3???105??或