Question

如图所示,AD是△ABC的角平分线,EF垂直平分AD,分别交AB于E,交AC于F,则四边形AEDF是菱形吗?

Answer 1

∵∠DAE=∠DAF,∠AOE=∠AOF=90°,

∠AEF=180°-∠DAE-∠AOE,∠AFE=180°-∠DAF-∠AOF,

∴∠AEF=∠AFE

∴△AEF是等腰三角形,AE=AF.

∵在直角△AOF和直角△DOF中,AF²=AO²+OF²,DF²=DO²+OF²,AO=DO,

∴AF=DF,

同理,AE=DE,

综上所述,AE=DE=AF=DF,四边形AEDF是菱形.