如图,AD//BC,AE,BE分别为DAB,ABC的平分线,点E在CD上,求证:AB=BC+AD.

如图,AD//BC,AE,BE分别为DAB,ABC的平分线,点E在CD上,求证:AB=BC+AD.
如图,AD//BC,AE,BE分别为DAB,ABC的平分线,点E在CD上,求证:AB=BC+AD.

如图,AD//BC,AE,BE分别为DAB,ABC的平分线,点E在CD上,求证:AB=BC+AD.
过E做AD的平行线交AB于F,则EF也平行于BC,∠DAE=∠AEF=∠EAF,所以三角形AEF为等腰三角形,所以AF=EF.同理,BF=EF,所以AF=EF=BF,又因为AD//BC//EF,所以DE=CE.在梯形ABCD中,所以有AD+BC=2EF,又AB=AF+EF=2EF.所以AB=BC+AD.得证!