矩形ABCD的内接平行四边形EFGH的各边与矩形的两条对角线分别平行.求证平行四边形EFGH的周长为定值

Question

矩形ABCD的内接平行四边形EFGH的各边与矩形的两条对角线分别平行.求证平行四边形EFGH的周长为定值

Answer 1

设EF=a,AC=b,EH=c
            ∵矩形ABCD
            ∴AC=BD=b  
            ∵EF‖AC
           ∴△EBF∽△ABC
         ∴EF:AC=BE:AB
         ∴a：b=BE:AB
          ∴aAB=bBE
         ∴a=bBE/AB 
       同理
      ∴△AEH∽△ABD
         ∴AE:AB=EH:BD
           AE:AB=c:b
          ∴cAB=bAE
            c=bAE/AB 
           ∵平行四边形EFGH
         ∴EH=FG,EF=GH
         ∴周长=2EF+2EH
                   =2a+2c
                  = 2(a+c)
                  =2(bEB/AB+bAE/AB)
                 =2b[(EB+AE)/AB]
                  =2b           
           有图,不知道图对不对,自己按照意思画的,方法对的,应该看的懂吧,图不对再改下啊,呵呵 ,那些算式最好把它抄下来看的清楚些