Consider a square $ABCD$ with center $O$. Let $E, F, G$, and $H$ be points on the interiors of sides $AB, BC, CD$, and $DA$, respectively, such that $AE = BF = CG = DH$. It is known that $OA$ intersects $HE$ at point $X$, $OB$ intersects $EF$ at point $Y$, $OC$ intersects $FG$ at point $Z$, and $OD$ intersects $GH$ at point $W$. Let $x$ and $y$ be the lengths of $AE$ and $AH$, respectively.

a) Given that Area $(EFGH) = 1 \text{ cm}^2$, calculate the value of $x^2 + y^2$.

b) Verify that $HX = \frac{y}{x+y}$. Then, conclude that $X, Y, Z$, and $W$ are vertices of a square.

c) Calculate

Area $(ABCD) \cdot$ Area $(XYZW)$.