Let $ABCD$ be a square and $O$ is your center. Let $E,F,G,H$ points in the segments $AB,BC,CD,AD$ respectively, such that $AE = BF = CG = DH$. The line $OA$ intersects the segment $EH$ in the point $X$, $OB$ intersects $EF$ in the point $Y$, $OC$ intersects $FG$ in the point $Z$ and $OD$ intersects $HG$ in the point $W$. If the $(EFGH) = 1$. Find:
$(ABCD) \times (XYZW)$
Note $(P)$ denote the area of the polygon $P$.
1. **Identify the given information and setup the problem:**
   - \(ABCD\) is a square with center \(O\).
   - Points \(E, F, G, H\) are on segments \(AB, BC, CD, AD\) respectively such that \(AE = BF = CG = DH\).
   - The line \(OA\) intersects segment \(EH\) at point \(X\).
   - The line \(OB\) intersects segment \(EF\) at point \(Y\).
   - The line \(OC\) intersects segment \(FG\) at point \(Z\).
   - The line \(OD\) intersects segment \(HG\) at point \(W\).
   - The area of quadrilateral \(EFGH\) is given as 1.

2. **Determine the side length of the square \(ABCD\):**
   - Let the side length of the square \(ABCD\) be \(s\).
   - Since \(E, F, G, H\) are equally spaced on the sides of the square, \(AE = BF = CG = DH = k\), where \(0 < k < s\).

3. **Calculate the coordinates of points \(E, F, G, H\):**
   - Assume the square is aligned with the coordinate axes with \(A = (0, 0)\), \(B = (s, 0)\), \(C = (s, s)\), and \(D = (0, s)\).
   - Then, \(E = (k, 0)\), \(F = (s, k)\), \(G = (s-k, s)\), and \(H = (0, s-k)\).

4. **Find the coordinates of the center \(O\):**
   - The center \(O\) of the square is \(\left(\frac{s}{2}, \frac{s}{2}\right)\).

5. **Determine the coordinates of points \(X, Y, Z, W\):**
   - The line \(OA\) intersects \(EH\) at \(X\). Since \(O\) is the midpoint of the square, the line \(OA\) is a diagonal. The intersection point \(X\) will be at the midpoint of \(EH\).
   - Similarly, \(Y\) is the midpoint of \(EF\), \(Z\) is the midpoint of \(FG\), and \(W\) is the midpoint of \(HG\).

6. **Calculate the area of quadrilateral \(XYZW\):**
   - Since \(X, Y, Z, W\) are midpoints of the segments \(EH, EF, FG, HG\), the quadrilateral \(XYZW\) is similar to \(EFGH\) but scaled down by a factor of \(\frac{1}{2}\).
   - The area of \(XYZW\) is \(\left(\frac{1}{2}\right)^2\) times the area of \(EFGH\).

7. **Calculate the area of \(ABCD\):**
   - The area of the square \(ABCD\) is \(s^2\).

8. **Calculate the product of the areas:**
   - Given that the area of \(EFGH\) is 1, the area of \(XYZW\) is \(\left(\frac{1}{2}\right)^2 \times 1 = \frac{1}{4}\).
   - Therefore, the product of the areas of \(ABCD\) and \(XYZW\) is \(s^2 \times \frac{1}{4}\).

9. **Determine the value of \(s\):**
   - Since the area of \(EFGH\) is 1, and it is a smaller square inside \(ABCD\), we can infer that \(s = 2\).

10. **Final calculation:**
    \[
    (ABCD) \times (XYZW) = s^2 \times \frac{1}{4} = 4 \times \frac{1}{4} = 1
    \]

The final answer is \(\boxed{1}\).