3、如图,在△ABC中,AB=AC,AD⊥BC于D,点P在BC上,PE⊥BC交BA延长线于E,交AC于F.求证：2AD=PE＋PF

题目

3、如图,在△ABC中,AB=AC,AD⊥BC于D,点P在BC上,PE⊥BC交BA延长线于E,交AC于F.求证：2AD=PE＋PF
A
P
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B
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B
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B

答案

证明：过点A作AG⊥PE交PE于G∵AB=AC,AD⊥BC∴∠BAB=∠CAD∵AD⊥BC,PE⊥BC,AG⊥PE∴矩形ADPG∴AD=PG,AD∥PE∴∠E=∠BAD,∠AFE=∠CAD∴∠E=∠AFE∴AE=AF∵AG⊥PE∴EG=FG∵PE=PG+EG,PF=PG-FG∴PE+PF=PG+EG+PG-FG=2PG∴2...