△abc中,ab=ad,ad是△abc外角的平分线,已知∠bac=∠acd.(1)求证:△abc≌△cda; (2)若∠b=60°,求证:四边形abcd是菱形.

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△abc中,ab=ad,ad是△abc外角的平分线,已知∠bac=∠acd.(1)求证:△abc≌△cda;(2)若∠b=60°,求证:四边形abcd是菱形.△abc中,ab=ad,ad是△abc外角

△abc中,ab=ad,ad是△abc外角的平分线,已知∠bac=∠acd.(1)求证:△abc≌△cda; (2)若∠b=60°,求证:四边形abcd是菱形.
△abc中,ab=ad,ad是△abc外角的平分线,已知∠bac=∠acd.
(1)求证:△abc≌△cda; (2)若∠b=60°,求证:四边形abcd是菱形.

△abc中,ab=ad,ad是△abc外角的平分线,已知∠bac=∠acd.(1)求证:△abc≌△cda; (2)若∠b=60°,求证:四边形abcd是菱形.
证明:(1)∵AB=AC,
∴∠B=∠ACB,
∵∠FAC=∠B+∠ACB=2∠ACB,
∵AD平分∠FAC,
∴∠FAC=2∠CAD,
∴∠CAD=∠ACB,
∵在△ABC和△CDA中
∠BAC=∠DCA AC=AC ∠DAC=∠ACB ,
∴△ABC≌△CDA;
(2)∵∠FAC=2∠ACB,∠FAC=2∠DAC,
∴∠DAC=∠ACB,
∴AD∥BC,
∵∠BAC=∠ACD,
∴AB∥CD,
∴四边形ABCD是平行四边形,
∵∠B=60°,AB=AC,
∴△ABC是等边三角形,
∴AB=BC,
∴平行四边形ABCD是菱形.