There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points?
1. **Understanding the Problem:**
   We are given 2017 distinct points in the plane and need to find the minimum number of distinct midpoints formed by all possible pairs of these points.

2. **Total Number of Pairs:**
   The total number of pairs of points from 2017 points is given by the combination formula:
   \[
   \binom{2017}{2} = \frac{2017 \times 2016}{2} = 2033136
   \]
   However, we are interested in the minimum number of distinct midpoints, not the total number of pairs.

3. **Optimal Configuration:**
   To minimize the number of distinct midpoints, we consider placing the points in a specific configuration. One effective way is to place all 2017 points equidistantly on a straight line.

4. **Midpoints on a Line:**
   When points are placed on a straight line, the midpoint of any two points \(A(x_i)\) and \(B(x_j)\) is given by:
   \[
   M_{ij} = \left( \frac{x_i + x_j}{2}, 0 \right)
   \]
   where \(x_i\) and \(x_j\) are the coordinates of points \(A\) and \(B\) respectively.

5. **Distinct Midpoints Calculation:**
   If the points are placed equidistantly on the \(x\)-axis, say at coordinates \(x_1 < x_2 < \ldots < x_{2017}\), the midpoints will be:
   \[
   \frac{x_1 + x_2}{2}, \frac{x_1 + x_3}{2}, \ldots, \frac{x_1 + x_{2017}}{2}, \frac{x_2 + x_3}{2}, \ldots, \frac{x_{2016} + x_{2017}}{2}
   \]
   These midpoints are distinct because each pair \((x_i, x_j)\) where \(i < j\) produces a unique value.

6. **Counting Distinct Midpoints:**
   The number of distinct midpoints is the number of unique pairs of points, which is:
   \[
   \binom{2017}{2} = 2017 \times 1008 = 2033136
   \]
   However, this is the total number of midpoints, not the minimum distinct midpoints. 

7. **Minimum Distinct Midpoints:**
   By placing the points equidistantly on a line, the midpoints will be distinct and the minimum number of distinct midpoints is:
   \[
   2017 - 1 = 2016
   \]
   This is because each midpoint is formed by a unique pair of points.

8. **Verification:**
   To verify, consider the midpoints formed by the points \(x_1, x_2, \ldots, x_{2017}\). The midpoints will be:
   \[
   \frac{x_1 + x_2}{2}, \frac{x_1 + x_3}{2}, \ldots, \frac{x_1 + x_{2017}}{2}, \frac{x_2 + x_3}{2}, \ldots, \frac{x_{2016} + x_{2017}}{2}
   \]
   These midpoints are distinct and the minimum number of distinct midpoints is indeed 2016.

The final answer is \(\boxed{2016}\).