Question 189
Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it.
Call a pair of points of the polyhedron antipodal if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes.
Let $A$ be the number of antipodal pairs of vertices, and let $B$ be the number of antipodal pairs of midpoints of edges. Determine the difference $A-B$ in terms of the numbers of vertices, edges and faces.
Call a pair of points of the polyhedron antipodal if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes.
Let $A$ be the number of antipodal pairs of vertices, and let $B$ be the number of antipodal pairs of midpoints of edges. Determine the difference $A-B$ in terms of the numbers of vertices, edges and faces.