Rt△ABC中∠BAC=90°,AD⊥BC于D,BG平分∠ABC,EF∥BC且交AC于F,求证AE=CF

问题描述:

Rt△ABC中∠BAC=90°,AD⊥BC于D,BG平分∠ABC,EF∥BC且交AC于F,求证AE=CF

证明:过E点作AC的平行线,交AB于P,交BC于Q∵∠BAC=90°,且PQ平行AC∴∠EPB=90°∴∠PAE+∠PEA=90°.∵AD⊥BC∴∠DEQ+∠EQD=90°∵∠PAE=∠DEQ∴∠PEA=∠EQD且BG是∠ABC的平分线,所以∠ABE=∠QBE∴△BEA≌△BEQ∴AE=E...