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The triangles  AOB and EOF are similar.    =>  EF = AB / 2
the triangles  FOG and BOC are similar.  =>  FG = BC/2
the triangles  GOH and  COD are similar.  =>  GH = CD / 2
the triangles   HOE and   DOA are similar.  =>  HE = DA / 2

So the perimeter ratio =  (EF+FG+GH+HE)/ (AB+BC+CD+DA)
              = 1/2

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