Question

[Lê Song Phương]

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 \(\frac{YZ}{BC}\)

Answer 1

[Xyz OLM]

I) Hình bạn tự vẽ nha 

Ta có DY//BH ; YH//DB 

=> DYHB hình bình hành => DY = HB 

Tương tự được ZE = FC

mà \(\frac{BH}{BC}=1-\frac{HC}{BC}=1-\frac{1}{\sqrt{2}}\)\(\left(\Delta HIC\approx\Delta BAC;\frac{AB}{IH}=\sqrt{2}\right)\)(1)

Tương tự được \(\frac{FC}{BC}=1-\frac{BF}{BC}=1-\frac{1}{\sqrt{2}}\)(2) 

Từ (1) ; (2) => BH = FC hay DY = ZE