Given a triangle ABC. D is a point on AB and E is a point on AC so that DE//BC and \(\frac{BC}{DE}=\sqrt{2}\), F is a point on BC and G is a point on AB so that FG//AC and \(\frac{AC}{FG}=\sqrt{2}\), H is a point on BC and I is a point on AC so that HI//AB and \(\frac{AB}{HI}=\sqrt{2}\). FG meets HI at X, DE meets HI at Y and DE meets FG at Z.
i) Prove that \(DY=ZE\)
ii) Find the exactly value of the ratio \(\frac{YZ}{BC}\)