Question

∩ACB=90°，∩BAC=30°，分别以AB、AC为边在△ABC的外侧作等边△ABE和等边△ACD

求证：EF=FD

Answer 1

证明:

如图,DG⊥AC,与AC相交于G,与AB相交H

设GH=a

则AH=2GH=2a,

DH=2AH=4a...①

∴GD=DH-GH=3a

∴AC=CD=2√(3)a

∴AE=AB=AC•cos30°=4a...②

由①②得DH=AE...③

∵DG⊥AC,AE⊥AC

∴DG∥AE

∴∠AEF=∠HDF...④

同时∠HFD=∠AFE..⑤

由③④⑤得△AFE≌HFD

∴EF=FD