Let be given a tetrahedron whose any two opposite edges are equal. A plane varies so that its intersection with the tetrahedron is a quadrilateral. Find the positions of the plane for which the perimeter of this quadrilateral is minimum, and find the locus of the centroid for those quadrilaterals with the minimum perimeter.
1. **Identify the given conditions and notation:**
   - We have a tetrahedron with opposite edges equal.
   - Let the lengths of the three pairs of opposite edges be \(a \leq b \leq c\).
   - We need to find the positions of the plane for which the intersection with the tetrahedron is a quadrilateral with minimum perimeter.
   - We also need to find the locus of the centroid of these quadrilaterals.

2. **Understand the geometry of the tetrahedron:**
   - A tetrahedron has four vertices and six edges.
   - Opposite edges in a tetrahedron do not share a common vertex.
   - Given that opposite edges are equal, we can denote the pairs of opposite edges as \((a, a)\), \((b, b)\), and \((c, c)\).

3. **Consider the intersection of the plane with the tetrahedron:**
   - The intersection of a plane with a tetrahedron can be a triangle or a quadrilateral.
   - For the intersection to be a quadrilateral, the plane must intersect all four faces of the tetrahedron.

4. **Determine the minimal perimeter condition:**
   - To minimize the perimeter of the quadrilateral, the plane should intersect the tetrahedron in such a way that the resulting quadrilateral has the shortest possible edge lengths.
   - This occurs when the plane is parallel to the pair of edges with the smallest length, \(a\).

5. **Find the locus of the centroid:**
   - The centroid of a quadrilateral is the average of its vertices.
   - When the plane is parallel to the pair of edges with length \(a\), the intersection quadrilateral will have its centroid lying on the line segment connecting the midpoints of these edges.
   - This line segment is denoted as \(\ell\).

6. **Conclusion:**
   - The plane that minimizes the perimeter of the quadrilateral is parallel to the pair of edges with length \(a\).
   - The locus of the centroid of these minimal quadrilaterals is the line segment \(\ell\) connecting the midpoints of the pair of edges with length \(a\).

The final answer is \( \boxed{ \ell } \) is the locus of the centroid of these minimal quadrilaterals.