Let $(O)$ denote a circle with a chord $AB$, and let $W$ be the midpoint of the minor arc $AB$. Let $C$ stand for an arbitrary point on the major arc $AB$. The tangent to the circle $(O)$ at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$, respectively. The lines $W X$ and $W Y$ meet $AB$ at points $N$ and $M$ , respectively. Does the length of segment $NM$ depend on position of $C$ ?
1. **Define the Circle and Points:**
   - Let the circle \((O)\) have center \(O(0,0)\) and radius \(r\).
   - Let \(A(-a, -b)\) and \(B(a, -b)\) be points on the circle such that \(a^2 + b^2 = r^2\).
   - Let \(W(0, -r)\) be the midpoint of the minor arc \(AB\).
   - Let \(C(r \cos \alpha, r \sin \alpha)\) be an arbitrary point on the major arc \(AB\).

2. **Find the Tangent Lines:**
   - The tangent line at \(C\) is given by:
     \[
     x \cos \alpha + y \sin \alpha = r
     \]
   - The tangent line at \(A\) is:
     \[
     ax - by = r^2
     \]
   - The tangent line at \(B\) is:
     \[
     ax + by = r^2
     \]

3. **Find Intersection Points \(X\) and \(Y\):**
   - The intersection point \(X\) of the tangent at \(C\) and the tangent at \(A\) is:
     \[
     X \left( \frac{r(b + r \sin \alpha)}{b \cos \alpha + a \sin \alpha}, \frac{r(a - r \cos \alpha)}{b \cos \alpha + a \sin \alpha} \right)
     \]
   - The intersection point \(Y\) of the tangent at \(C\) and the tangent at \(B\) is:
     \[
     Y \left( \frac{r(b + r \sin \alpha)}{b \cos \alpha - a \sin \alpha}, \frac{r(-a - r \cos \alpha)}{b \cos \alpha - a \sin \alpha} \right)
     \]

4. **Find Intersection Points \(N\) and \(M\):**
   - The line \(WX\) intersects \(AB\) at point \(N\):
     \[
     N \left( \frac{(r - b)(b + r \sin \alpha)}{a(1 + \sin \alpha) + (b - r) \cos \alpha}, -b \right)
     \]
   - The line \(WY\) intersects \(AB\) at point \(M\):
     \[
     M \left( \frac{(r - b)(b + r \sin \alpha)}{-a(1 + \sin \alpha) + (b - r) \cos \alpha}, -b \right)
     \]

5. **Calculate the Distance \(NM\):**
   - The distance \(NM\) is the difference in the \(x\)-coordinates of \(N\) and \(M\):
     \[
     NM = x_N - x_M
     \]
   - Substituting the coordinates of \(N\) and \(M\):
     \[
     NM = \frac{(r - b)(b + r \sin \alpha)}{a(1 + \sin \alpha) + (b - r) \cos \alpha} - \frac{(r - b)(b + r \sin \alpha)}{-a(1 + \sin \alpha) + (b - r) \cos \alpha}
     \]
   - Simplifying the expression, we find:
     \[
     NM = a = \frac{1}{2} AB
     \]

Thus, the length of segment \(NM\) does not depend on the position of \(C\).

The final answer is \(\boxed{a = \frac{1}{2} AB}\)