$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$
1. **Identify the midpoints of segments \(AB\) and \(CD\):**
   Let \(P\) be the midpoint of segment \(AB\) and \(Q\) be the midpoint of segment \(CD\). Therefore, we have:
   \[
   P = \frac{A + B}{2} \quad \text{and} \quad Q = \frac{C + D}{2}
   \]

2. **Express the distances \(MA\), \(MB\), \(MC\), and \(MD\) using the coordinates of \(M\):**
   Let \(M\) be a point in space with coordinates \((x, y, z)\). Then the squared distances from \(M\) to points \(A\), \(B\), \(C\), and \(D\) are given by:
   \[
   MA^2 = (x - x_A)^2 + (y - y_A)^2 + (z - z_A)^2
   \]
   \[
   MB^2 = (x - x_B)^2 + (y - y_B)^2 + (z - z_B)^2
   \]
   \[
   MC^2 = (x - x_C)^2 + (y - y_C)^2 + (z - z_C)^2
   \]
   \[
   MD^2 = (x - x_D)^2 + (y - y_D)^2 + (z - z_D)^2
   \]

3. **Set up the given equation:**
   According to the problem, we need to find the locus of points \(M\) such that:
   \[
   MA^2 + MB^2 = MC^2 + MD^2
   \]

4. **Use the midpoint property:**
   By the properties of midpoints, we know:
   \[
   MA^2 + MB^2 = 2MP^2 + \frac{AB^2}{2}
   \]
   \[
   MC^2 + MD^2 = 2MQ^2 + \frac{CD^2}{2}
   \]
   where \(MP\) and \(MQ\) are the distances from \(M\) to the midpoints \(P\) and \(Q\) respectively.

5. **Simplify the equation:**
   Substituting these into the given equation, we get:
   \[
   2MP^2 + \frac{AB^2}{2} = 2MQ^2 + \frac{CD^2}{2}
   \]
   Simplifying further, we obtain:
   \[
   MP^2 - MQ^2 = \frac{CD^2 - AB^2}{4}
   \]

6. **Interpret the result geometrically:**
   The equation \(MP^2 - MQ^2 = \text{constant}\) represents a plane that is the perpendicular bisector of the line segment \(PQ\). This plane is equidistant from \(P\) and \(Q\).

Therefore, the locus of points \(M\) such that \(MA^2 + MB^2 = MC^2 + MD^2\) is the plane that is the perpendicular bisector of the line segment joining the midpoints \(P\) and \(Q\) of segments \(AB\) and \(CD\).

\(\blacksquare\)

The final answer is the plane that is the perpendicular bisector of the line segment joining the midpoints \( \boxed{ P } \) and \(Q\) of segments \(AB\) and \(CD\).