三角形ABC中,角ACB=90度,且AE垂直于BD的延长线于E,又BD=2AE,BD是角ABC的平分线,求证:AC=BC

问答/446℃/2025-04-19 20:53:10

优质解答：

延长AE和BC交于F∵BD平分∠ABC∴∠ABE=∠FBE∵BD⊥AE(BE⊥AF)∴∠BEA=∠BEF=90°∵BE=BE∴△ABE≌△FBE(ASA)∴AE=EF即AF=2AE∵BD=2AE∴AF=BD∵∠BDC=∠ADE∠AED=∠BCD=90°∴∠DAE=90°-∠ADE∠CBD=90°-∠BDC∴∠DAE=∠CBD即∠CAF=∠CBD∵∠ACF=∠BCD=90°∴△BCD≌△ACF(AAS)∴AC=BC

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