(本题10分)如图,在平行四边形$ABCD$中,$E$为$AB$边上的中点,连结$DE$并延长,交$CB$的延长线于点$F$.(1)求证:$AD=BF$;(2)若平行四边形$ABCD$的面积为32,试求四边形$EBCD$的面积.<ImageHere>
解：(1)证明：$because$点$E$是$AB$中点,$thereforeAE=BE$,又$because$四边形$ABCD$是平行四边形,$thereforeAD//BC$,又$because$点$F$在$CB,DE$延长线上,$thereforeAD//BF$,$thereforeangleADE=angleBFE$,在$triangleAED$与$triangleBEF$中,$left{begin{array}{l}angleADE=angleBFE,angleAED=angleBEF,AE=BE,end{array}right.$$thereforetriangleAEDcongtriangleBEF(AAS)$,$thereforeAD=BF$(2)$becauseEB//CD$,$thereforetriangleEFBsimtriangleFDC$,$becausetriangleAEDcongtriangleBEF$,$thereforeED=EF,quadS_{triangleAED}=S_{triangleBEF}$,$becausefrac{EF}{DF}=frac{1}{2}$,$thereforefrac{S_{triangleBEF}}{S_{triangleDCF}}=frac{1}{4}$,$therefore$设$S_{triangleBFE}$为$x$,则$S_{text{四边形}EBCD}$为$3x$,由$4x=32$,得$x=8$,$thereforeS_{text{四边形}EBCD}=3times8=24$.