Question

如图,在矩形ABCD中,F是CD的中点,在BC上取一点E,使AF平分∠DAE,又AE=DC+CE,求证：四边形ABCD是

Answer 1

延长AF,交BC的延长线于点G.

在△ADF和△GCF中,∠AFD = ∠GFC ,DF = CF ,∠ADF = 90°= ∠GCF ,

所以,△ADF ≌ △GCF ,可得：AD = GC ,∠DAF = ∠CGF .

∠EAF = ∠DAF = ∠CGF ,可得：EG = AE ；

则有：DC = AE-CE = EG-CE = GC = AD ,

所以,四边形ABCD是正方形.（邻边相等的矩形是正方形）