如图,四边形ABCD是菱形,AF⊥BC交BD于E,交BC于F.求证：AD^2=1/2DE×DB.

如图,四边形ABCD是菱形,AF⊥BC交BD于E,交BC于F.求证：AD^2=1/2DE×DB.
如图,四边形ABCD是菱形,AF⊥BC交BD于E,交BC于F.求证：AD^2=1/2DE×DB.

如图,四边形ABCD是菱形,AF⊥BC交BD于E,交BC于F.求证：AD^2=1/2DE×DB.
取DE中点H,连AH,因∠EAD=90°,则DH=AH
∴∠HAD=∠ADH
∵四边形ABCD是菱形
∴∠ABD=∠CBD
∵∠ADH=∠CBD
∴∠DAH=∠ABD
∵∠ADB=∠HDA
∴△AHD∽△BAD
∴AD:DH=DB:AD
∴AD^2=DH*DB
∵DH=1/2DE
∴AD^2=1/2DE×DB.