Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.
1. **Identify the problem and the goal:**
   We are given \( n \geq 2 \) points inside a closed unit disc and need to evaluate the maximum product of the lengths of all \( \frac{n(n-1)}{2} \) segments determined by these \( n \) points.

2. **State the proposed solution:**
   The proposed solution claims that the maximum product \( P \) is \( n^{n/2} \), achieved when the points are the vertices of an inscribed regular \( n \)-gon.

3. **Set up the problem using complex coordinates:**
   Let the points have complex coordinates \( a_1, a_2, \ldots, a_n \) with \( |a_k| \leq 1 \) for all \( k \). Using the Vandermonde determinant, we have:
   \[
   P = \prod_{i < j} |a_i - a_j| = |\det A|,
   \]
   where
   \[
   A = \begin{bmatrix}
   1 & a_1 & a_1^2 & \cdots & a_1^{n-1} \\
   1 & a_2 & a_2^2 & \cdots & a_2^{n-1} \\
   \vdots & \vdots & \vdots & \ddots & \vdots \\
   1 & a_n & a_n^2 & \cdots & a_n^{n-1}
   \end{bmatrix}.
   \]

4. **Bound the determinant:**
   Each entry of \( A \) has magnitude at most 1. We need to show that \( |\det A| \leq n^{n/2} \) for any \( n \times n \) matrix \( A \) with entries in the unit disk.

5. **Use properties of Hermitian matrices:**
   Let \( v_1, v_2, \ldots, v_n \) be the column vectors of \( A \). We aim to show \( |\det A| \leq |v_1| \cdots |v_n| \). By scaling each column, assume \( |v_1| = |v_2| = \cdots = |v_n| = 1 \). We need to show \( |\det A| \leq 1 \).

6. **Construct the matrix \( M \):**
   Let \( M = A^{\dagger}A \), where \( A^{\dagger} \) is the conjugate transpose of \( A \). \( M \) is Hermitian and positive semidefinite, meaning its eigenvalues \( \lambda_1, \lambda_2, \ldots, \lambda_n \) are nonnegative.

7. **Apply the AM-GM inequality:**
   Since \( \det M = \lambda_1 \lambda_2 \cdots \lambda_n \) and \( \operatorname{tr} M = \lambda_1 + \lambda_2 + \cdots + \lambda_n = n \), we have:
   \[
   |\det A|^2 = |\det M| = |\lambda_1 \lambda_2 \cdots \lambda_n| \leq \left( \frac{\lambda_1 + \lambda_2 + \cdots + \lambda_n}{n} \right)^n = 1.
   \]
   Therefore, \( |\det A| \leq 1 \).

8. **Conclusion:**
   Since \( |\det A| \leq 1 \) and the product \( P \) is maximized when the points are vertices of an inscribed regular \( n \)-gon, the maximum product is \( n^{n/2} \).

The final answer is \( \boxed{ n^{n/2} } \).