Question 33
Consider a square $ABCD$ with center $O$. Points $E,F,G,H$ lie in the interiors of sides $AB,BC,CD,DA$, respectively, with $AE=BF=CG=DH$. The segments $OA,OB,OC,OD$ meet $HE,EF,FG,GH$ at $X,Y,Z,W$, respectively. Let $x=AE$ and $y=AH$.
a) Given $\operatorname{Area}(EFGH)=1\ \mathrm{cm}^2$, find $x^2+y^2$.
b) Verify that $HX=\frac{y}{x+y}$. Then show that $X,Y,Z,W$ are the vertices of a square.
c) Calculate $\operatorname{Area}(ABCD)\cdot\operatorname{Area}(XYZW)$.
a) Given $\operatorname{Area}(EFGH)=1\ \mathrm{cm}^2$, find $x^2+y^2$.
b) Verify that $HX=\frac{y}{x+y}$. Then show that $X,Y,Z,W$ are the vertices of a square.
c) Calculate $\operatorname{Area}(ABCD)\cdot\operatorname{Area}(XYZW)$.