Question

两圆相交于点A和点B,由点A作两圆的切线,分别与两圆相交于M和N,直线BM和BN分别与两圆交与点P、点Q,求证MP=NQ

Answer 1

连接AP、AB、AQ、MQ

∵AN、AM是切线

∴∠NAB=∠AMB,∠MAB=∠ANB

∴∠ABQ=∠ANB+∠NAB=∠AMB+∠MAB

∴∠AMQ=∠ABQ=∠AMB+∠MAB

∵∠AQB=∠AMB.∠MQB=∠MAB

∴∠AQM=∠AQB+∠MQB=∠AMB+∠MAB

∴∠AMQ=∠AQM

∴AM=AQ

∵∠APB=∠ANB即∠APM=∠ANQ

∠AGB=∠AMB即∠AQN=∠AMP

∴△APM≌△ANQ(AAS)

∴MP=NQ